UCNRLf 


B  3  laa  c,j_j_ 


Edward  Bright; 


MaiiLematJjCLa--I)ei:it-.-- 


THE  SCIENCE  OF  MECHANICS 

(SUPPLEMENTARY   VOLUME) 


-Uixa^    fl^^;^^ 


THE    SCIENCE   OF 
MECHANICS 

A  CRITICAL  AND  HISTORICAL 
ACCOUNT  OF  ITS  DEVELOPMENT 

BY 

ERNST    MACH 

EMERITUS   PROFESSOR    OF    THE    HISTORY   AND   THEORY   OF 
INDUCTIVE    SCIEN'CE    IN   THE    UNIVERSITY   OF    VIENNA 


SUPPLEMENT    TO    THE 
THIRD    ENGLISH    EDITION 

^    CONTAINING  THE  AUTHOR'S  ADDITIONS  TO  THE 
SEVENTH  GERMAN  EDITION 

TRANSLATED   AND    ANNOTATED   BY 

PHILIP  E.  B.  JOURDAIN 

M.  A.  (Cantab.) 


CHICAGO  AND   LONDON 
THE  OPEN  COURT  PUBLISHING  COMPANY 

1915 


Copyright  in  Great  Britain  under  the  Act  0/  igii 


PUBLISHERS'  PREFACE  TO  THE 
SUPPLEMENTARY  VOLUME 

The  first  edition  of  the  English  translation  by 
Mr  M'Cormack  of  Mach's  Mechanics  was  published 
in  1893,  ^i"^d  was  carefully  revised  by  Professor 
Mach'  himself.  Since  then  two  other  editions  of 
this  translation  have  appeared,  in  which  the  altera- 
tions contained  in  the  successive  German  editions 
have  been  embodied  in  the  form  of  appendices. 
In  the  seventh  German  edition,  however,  which 
appeared  at  Leipsic  (F.  A.  Brockhaus)  in  1912,  there 
have  been  more  profound  modifications  in  the  plan 
of  Professor  Mach's  work,  which  are  shortly  referred 
to  in  the  preface  to  that  edition.  Many  things  are 
added  and  some  things  are  omitted.  Among  the 
parts  omitted  are  the  prefaces  to  all  of  the  German 
editions  except  the  first,  and  a  new  preface  to 
the  seventh  edition  has  been  added.  The  most 
extensive  additions  relate  to  recent  historical  re- 
searches on  the  work  of  Galileo's  precursors  and 
the  early  work  of  Galileo  himself ;  and  the  book 
is  dedicated  to  the  late  Emil  Wohlwill,  of  whose 
researches  much  use  has  been  made. 

V 


?8i.507 


vi  THE  SCIENCE  OF  MECHANICS 

In  the  present  English  edition,  after  much  thought 
and  consultation  with  Professor  Mach  and  at  the 
suggestion  of  Mr  Philip  E.  B.  Jourdain,  we  have 
adopted  a  different  plan.  Mr  Jourdain  has  assumed 
the  responsibility  of  a  revision  of  the  Mechanics  on 
the  basis  of  the  seventh  German  edition,  and  has 
signified  the  alterations  to  text  and  appendix  in  the 
appendix  printed  here.  The  only  other  addition  to 
the  seventh  German  edition  is  a  portrait  of  Newton 
after  Kneller's  well-known  picture.  This  very 
welcome  addition — no  portrait  of  the  greatest  of 
mechanical  inquirers  having  adorned  previous 
editions  of  the  Mechanics — is  also  given  as  the 
frontispiece  of  the  present  volume.  The  reader 
who  possesses  the  third  English  edition  of  the 
Mechanics  ^  as  well  as  this  volume  has  a  complete 
picture  of  the  various  stages  through  which  Mach's 
Mechanics  has  passed. 

And  this  retention  of  the  successive  alterations 
and  additions  seems  almost  to  be  necessary.  Indeed, 
Mach's  work  is  to  be  regarded  not  only  as  a  con- 
tribution to  the  enlightenment  of  so  many  points  in 
the  history  and  the  principles  of  mechanics,  but  also 
as  a  foundation-stone  of  science,  which  is  of  the 
greatest  historical  interest  in  itself.  The  slow  but 
sure  progress  of  digestion  of  Mach's  ideas — which 


*  The  Science  of  Mechanics  :  A  Critical  and  Historical  Account  of 
its  Develop?)ienf ,  translated  by  T.  J.  M'Cormack,  third  edition  ; 
Chicago  and  London  :   The  Open  Court  Publishing  Company,  1907. 


PUBLISHERS'  PREFACE  vii 

must  have  seemed  so  revolutionary  to  most  of  our 
modern  schoolmen — and  the  ever-growing  influence 
of  these  ideas  on  teaching  are  both  reflected  in 
these  prefaces.  And  where  Mach's  historical  know- 
ledge has  grown — and  grown,  it  is  to  be  observed, 
in  consequence  of  the  researches  of  others  who 
were  often  inspired  by  Mach's  own  work, — it  is 
surely  of  absorbing  interest  still  to  be  able  to  read 
the  original  form  of  Mach's  work  and  compare  it 
with  the  later  emendations. 

At  the  end  of  these  appendices,  Mr  Jourdain  has 
added  'some  notes  of  his  own  which  Professor  Mach 
has  commended  in  his  preface  to  the  seventh  German 
edition.  Any  other  notes  which  have  been  added 
to  the  text  of  Mach's  appendix  for  the  purpose  of 
completing  or  correcting  references,  or  of  referring 
to  more  generally  accessible  editions  or  translations 
of  the  works  cited  by  the  author,  are  enclosed  in 
square  brackets. 

In  spite  of  many  obstacles  and  inconveniences, 
occasioned  mainly  by  the  inability  to  use  his  right 
hand,  Professor  Mach  has  most  kindly  revised  the 
entire  work  of  Mr  Jourdain,  including  all  additions 
and  alterations. 


AUTHOR'S  PREFACE  TO  THE 
SEVENTH  GERMAN  EDITION 

When,  forty  years  ago,  I  first  expressed  the 
ideas  explained  in  this  book,  they  found  small 
sympathy,  and  indeed  were  often  contradicted. 
Only  a  few  friends,  especially  Josef  Popper  the 
engineer,  were  actively  interested  in  these  thoughts 
and  encouraged  the  author.  When,  two  years 
later,  Kirchhoff  published  his  well-known  and  often- 
quoted  dictum,  which  even  to-day  is  hardly  correctly 
interpreted  by  the  majority  of  physicists,  people 
liked  to  think  that  the  author  of  the  present  work 
had  misunderstood  Kirchhoff.  I  must  decline  with 
thanks  this,  as  it  were,  prophetical  misunderstand- 
ing as  not  corresponding  either  to  my  faculty  of 
presentiment  or  to  my  powers  of  understanding. 

However,  the  book  has  reached  a  seventh  German 
edition,  and  by  means  of  excellent  English,  French, 
Italian,  and  Russian  translations  has  spread  over 
almost  all  the  world.  Gradually  some  of  those  who 
work  at  this  subject,  like  J.  Cox,  Hertz,  Love, 
MacGregor,  Maggi,  H.  von  Seeliger,  and  others, 
gave  voice  to  their  agreement.  For  them,  of  course, 
only  details  in    a  book  meant  for  a  general  intro- 


X  THE  SCIENCE  OF  MECHANICS 

duction  could  be  of  interest.  In  this  subject,  I 
could  hardly  avoid  touching  upon  philosophical, 
historical,  and  epistemological  questions  ;  and  by 
this  the  attention  of  various  critics  was  aroused. 
I  took  special  joy  in  the  recognition  which  I  found 
with  the  philosophers  R.  Avenarius,  J.  Petzoldt, 
H.  Cornelius,  and,  later,  W.  Schuppe.  The  ap- 
parently small  concessions  which  philosophers  of 
another  tendency,  like  G.  Heymans,  P.  Natorp, 
and  Aloys  Miiller,  have  granted  to  my  characterisa- 
tion of  absolute  space  and  absolute  time  as  mis- 
conceptions suffice  for  me  ;  indeed,  I  do  not  wish 
for  anything  more.  I  thank  Messrs  L.  Lange  and 
J.  Petzoldt  not  only  for  their  agreement  in  certain 
details,  but  also  for  their  active  and  fruitful  collabora- 
tion. In  a  historical  respect,  the  criticisms  of 
Emil  Wohlwill,  whose  death,  I  regret  to  say,  has 
just  been  announced  to  me,  were  valuable  and 
enlightening  to  me,  especially  on  the  period  of 
Galileo's  youthful  work  ;  further,  critical  remarks 
of  P.  Duhem  and  G.  Vailati  have  also  been  valuable. 
I  am  very  grateful  to  Mr  Philip  E.  B.  Jourdain  of 
Cambridge  for  his  critical  notes  that  unfortunately, 
for  the  most  part,  came  too  late  for  inclusion  in 
this  edition,  which  was  already  nearly  finished. 
P.  Duhem,  O.  Holder,  G.  Vailati,  and  P.  Volkmann 
have  taken  part  in  the  epistemological  discussions 
with  vigour,  and  their  remarks  have  been  helpful 
to  me. 


AUTHOR'S  PREFACE  xi 

At  the  end  of  the  last  century  my  disquisitions 
on   mechanics  fared  well  as    a  rule  ;    it    may    have 
been  felt  that  the  empirico-critical  side  of  this  science 
was    the    most    neglected.      But    now    the  Kantian 
traditions  have  gained  power  once  more,  and  agam 
we    have    the  demand  for    an    a  priori   foundation 
of  mechanics.      Now,   I    am  indeed  of  the  opinion 
that  all  that  can  be  known  a  priori  of  an  empirical 
domain  must  become  evident  to  mere  logical  circum- 
spection only  after  frequent  surveys  of  this  domain, 
but  1  do  not  believe  that  investigations  like  those 
of  G.    HameP  do  any  harm  to  the  subject.      Both 
sides    of   mechanics,   the   empirical  and  the  logical 
side,    require    investigation.      I    think    that    this  is 
expressed  clearly  enough  in  my  book,  although  my 
work  is  for  good    reasons  turned  especially  to  the 

empirical  side. 

I  myself— seventy-four  years  old,  and  struck 
down  by  a  grave  malady— shall  not  cause  any  more 
revolutions.  But  I  hope  for  important  progress 
from  a  young  mathematician,  Dr  Hugo  Dingier, 
who,  judging  from  his  publications, ^  has  proved 
that'  he  has  attained  to  a  free  and  unprejudiced 
survey  of  both  sides  of  science. 

This  edition  will  be  found  somewhat  more  homo- 

1  «.fTK.r    R-^nm      Zeit    und    Kraft    als    apriorische    Formen     der 

vorx:t^9o{f  "Uber  c^:  Grundlagen  der  Mechan.U,"  Matk.  Ann., 

""^"^^^nzf^ndZiele  der  Wissenschaft,  1910;   D^c  Grundlagen  der 
angewandtcn  Geometric y  19  n. 


xii        THE  SCIENCE  OF  MECHANICS 

geneous  than  the  former  ones.  Many  an  ancient 
dispute  which  to-day  interests  nobody  any  more 
is  left  out  and  many  new  things  are  added.  The 
character  of  the  book  has  remained  the  same.  With 
respect  to  the  monstrous  conceptions  of  absolute 
space  and  absolute  time  I  can  retract  nothing.  Here 
I  have  only  shown  more  clearly  than  hitherto  that 
Newton  indeed  spoke  much  about  these  things,  but 
throughout  made  no  serious  application  of  them. 
His  fifth  corollary  ^  contains  the  only  practically 
usable  (probably  approximate)  inertial  system. 

ERNST  MACH. 
Vienna,  February  ^th,  1912. 

^  Frincipia,  1687,  p.  19, 


TABLE   OF   CONTENTS 


PAGE 


Publishers'     Preface    to     the     Supplementary 

Volume    v 

Author's    Preface    to     the    Seventh     German 

Edition    ........        ix 

Table  ^of  Contents        ......     xiii 

Appendix  of  Additions  and  Alterations  to  the 

Seventh  German  Edition       .         .         .         .         i 

I.  On  Archimedes'  demonstration  of  the  principle  of  the 
Lever,  i. — II.  Del  Monte  and  the  principle  of  virtual 
displacements,  3. — III.  Galileo  and  the  conception  of 
work,  4.  —  IV.  The  development  of  Statics  with  Jor- 
danus  Nemorarius,  an  anonymous  precursor  of  Leonardo 
da  Vinci,  Leonardo  da  Vinci,  Cardano,  Stevinus,  Des- 
cartes, and  others,  4. — V.  Voltaire's  ideas  on  the  air,  16. 
— VI.  Ancient  Dynamics,  17.  — VII.  Aristotle's  Dynam- 
ics, 17. — VIII.  The  ideas  of  Benedetti  and  the  early 
ideas  of  Galileo  on  projection,  18. — IX.  Galileo's  early 
speculations  on  Dynamics,  20. — X.  Galileo's  achieve- 
ments, and  his  precursors  and  contemporaries,  22. — 
XL  Baliani  and  the  conception  of  mass,  28. — XII.  New- 
ton's achievements  in  other  domains  of  Physics,  29, — 
XIII.  Galileo's  axes  of  reference  in  his  theory  of  the  tides, 
31. — XIV.  On  electromagnetic  mass,  31. — XV.  and 
XVI.  References,  31,  32. — XVII.  Relativity  of  mass,  32. 
— XVIII.  and  XIX.  Newton's  axes  of  reference,  33,  34. 
— XX.  Note  on  the  advocates  of  absolute  motion,  36. — 
XXI.  The  writings  on  the  law  of  inertia  of  C.  Neumann, 
Streintz,  L.  Lange,  Petzoldt,  and  others,  and  on  the 
works  of  modern  relativists,  36. — XXII.  The  tautology 
in  Newton's  enunciations,  46. — XXIII.  Mach's  defini- 

xiii 


xiv        THE  SCIENCE  OF  MECHANICS 


PAGE 


tion  of  mass,  46. — XXIV.  Thomson  and  Tait,  47.— 
XXV.  The  hypothetical  nature  of  our  dynamical  laws, 
47. — XXVI.  and  XXVII.  On  an  assumption  of  Galileo's, 
48. — XXVIII,  Hartmann  on  the  definition  of  force,  48. 
— XXIX.  References,  49.— XXX.  Some  alterations  in 
the  seventh  German  edition,  49. — XXXI.  Experiments 
with  radiometer  and  reaction-wheels,  ^o. — XXXII. 
Alterations  about  Marci,  51.— XXXIII.  On  the  prin- 
ciples of  Gauss  and  Ostwald,  51. — XXXIV.  Helm  and 
the  character  of  minimum  principles,  53. — XXXV.  Re- 
ferences on  the  principle  of  economy,  53. — XXXVI. 
Petzoldt  on  the  striving  for  stability  in  intellectual  life, 
54 — XXXVII.  Conclusion,  54. 

Corrections  to  be  made  in  the  Mechanics      .       58 

Notes   on    Mach's   Mechanics,  by   Philip   E.   B. 

jourdain  ........       63 

Index •         •     103 


THE    SCIENCE    OF 
MECHANICS 


APPENDIX  OF 
ADDITIONS  AND  ALTERATIONS  TO 
THE   SEVENTH   GERMAN  EDITION 

I 

[To  p.    515,   line  9  of  third  edition  of  Mechanics^ 
add:] 

I  must  here  draw  my  readers'  attention  to  a 
beautiful  paper  by  G.  Vailati,^  in  which  the  side 
of  Holder  against  my  criticism  of  Archimedes' 
deduction  of  the  law  of  the  lever  is  taken,  but  partly 
too  Holder  is  criticised.  I  believe  that  everyone 
may  read  Vailati's  exposition  with  profit  and,  by 
comparison  with  what  I  have  said  on  pp.  17-20 
of  the  third  edition  of  my  Mechanics,  will  be  in  a 
position  himself  to  form  a  judgment  upon  the 
points    at    issue.      Vailati    shows    that    Archimedes 

^  "La  dimostrazione  del  principio  delle  leva  data  da  Archimede," 
Bolletino  di  bibliograjia  e  storia  delle  scienze  matematiche,  May  and 
June  1904. 


2  THE  SCIENCE  OF  MECHANICS 

derives  the  law  of  the  lever  on  the  basis  of  general 
experiences  about  the  centre  of  gravity.  I  have 
never  disputed  the  view  that  such  a  process  is 
possible  and  permissible  and  even  very  fruitful  at 
a  certain  stage  of  investigation,  and  further,  is 
perhaps  the  only  correct  one  at  that  stage.  On  the 
contrary,  by  the  manner  in  which  I  have  exposed 
the  derivations  of  Stevinus  and  Galileo,  which  were 
made  after  the  example  of  Archimedes,  I  have 
expressly  recognised  this.  But  the  aim  of  my 
whole  book  is  to  convince  the  reader  that  we  cannot 
make  up  properties  of  nature  with  the  help  of  self- 
evident  suppositions,  but  that  these  suppositions 
must  be  taken  from  experience.  1  would  have  been 
false  to  this  aim  if  I  had  not  striven  to  disturb  the 
impression  that  the  general  law  of  the  lever  could 
be  deduced  from  the  equilibrium  of  equal  weights 
on  equal  arms.  I  had,  then,  to  show  where  the 
experience  that  already  contains  the  general  law  of 
the  lever  is  introduced.  Now  this  experience  lies 
in  the  supposition  emphasised  on  p.  14,  and  in  the 
same  way  it  lies  in  every  one  of  the  general  and 
undoubtedly  correct  theorems  on  the  centre  of 
gravity  brought  forward  by  Vailati.  Now,  because 
the  fact  that  the  value  of  a  load  is  proportional 
to  the  arms  of  the  lever  is  not  directly  and  in  the 
simplest  way  apparent  in  such  an  experience,  but 
is  found  in  an  artificial  and  roundabout  way,  and 
is  then  offered  to  the  surprised  reader,  the  modern 


ADDITIONS  AND  ALTERATIONS        3 

reader  has  to  object  to  the  deduction  of  Archimedes. 
This  deduction  from  simple  and  almost  self-evident 
theorems  may  charm  a  mathematician  who  either 
has  an  affection  for  Euclid's  method,  or  who  puts 
himself  into  the  appropriate  mood.  But  in  other 
moods  and  with  other  aims  we  have  all  the  reason 
in  the  world  to  distinguish  in  value  between  getting 
from  one  proposition  to  another  and  conviction,  and 
between  surprise  and  insight.  If  the  reader  has 
derived  some  usefulness  out  of  this  discussion,  I 
am  not  very  particular  about  maintaining  every 
word   I^  have  used. 

II 

[To  p.  49,  line  2,  add  :] 

In  my  exposition  in  the  preceding  editions, 
E.  Wohlwill  finds  that  the  achievements  of  Stevinus 
are  over-estimated  as  compared  with  those  of  del 
Monte  and  Galileo.  In  fact,  del  Monte,  in  his 
Mechanicorum  liber  (Pisauri,  1577),  considered  the 
lengths  of  the  paths  which  are  described  simultane- 
ously by  the  weights  in  the  cases  of  the  lever, 
pulleys,  and  wheel  and  axle.  His  consideration  is 
more  geometrical  than  mechanical.  Also,  with 
del  Monte  is  lacking  the  principle  by  which  the 
surprising  character  is  taken  away  from  the  effects 
of  machines  {cf.  Wohlwill,  Galilei^  i,  pp.  142 
et  seqq.).  Thus  del  Monte  was  out-distanced  by 
other   mediaeval  writers  who   concerned   themselves 


4  THE  SCIENCE  OF  MECHANICS 

with  the  heritage  of  the  principle  of  virtual  velocities 
which  had  been  handed  down  by  the  ancients, 
and  who  are  to  be  mentioned  on  another  occasion. 
Now,  at  the  end  of  the  sixteenth  century,  Stevinus 
did  not  advance  beyond  his  immediate  predecessor 
del  Monte. 

Ill 
[To  p.  52,  line  2,  add  :] 

E.  Wohlwill  emphasises  that  Galileo  laid  stress 
on  the  loss  of  velocity  which  corresponds  to  the 
economy  of  force  in  machines  {cf.  Galilei,  i, 
pp.  141,  142).  If  we  use  the  modern  conception — 
to  the  development  of  which  Galileo  contributed  so 
much — of  ''work,"  we  can  say  without  equivocal- 
ness  :  in  machines  work  is  not  economised. 

IV 

[To  p.  85,  last  line,  add  :] 

The  knowledge  of  the  development  of  a  science 
rests  on  the  study  of  writings  in  their  historical 
sequence  and  in  their  historical  connection.  For 
ancient  times  many  sources  are,  of  course,  lacking, 
and  for  other  times  the  author  is  unknown  or 
doubtful.  In  later  centuries,  especially  before  the 
discovery  of  printing,  the  bad  habit  is  general  of 
the  author  seldom  referring  to  his  predecessors 
where  he  uses  their  works,  and  usually  only  doing 
so    where    he    thinks    he    has    to    contradict    those 


ADDITIONS  AND  ALTERATIONS        5 

predecessors.  By  these  circumstances,  the  above 
study  is  made  very  difficult  and  makes  the  highest 
demands  on  criticism. 

P.  Duhem  develops  in  his  book,  Les  origines  de 
la  statique  (Paris,  1905,  vol.  i),  the  view  that 
E.  VVohlvvill  had  already  taken,  that  modern  scien- 
tific civilisation  is  much  more  intimately  connected 
with  ancient  scientific  civilisation  than  people 
usually  suppose.  The  scientific  thoughts  of  the 
Renascence  developed  very  slowly  and  gradually 
from  those  of  ancient  Greece,  particularly  from 
those  of  the  peripatetic  and  Alexandrian  school. 
I  will  here  emphasise  that  Duhem's  book  contains 
a  mine  of  stimulating,  instructive,  and  enlightening 
details  condensed  in  a  small  space.  To  the  know- 
ledge of  these  details  we  could  only  otherwise 
attain  by  a  wearisome  study  of  old  books  and 
manuscripts.  By  that  alone  the  reading  of 
Duhem's  work  excites  much  admiration  and  is 
very  fruitful. 

In  especial,  Duhem  ascribes  to  Jordanus  Nemor- 
arius,  a  writer  of  the  thirteenth  century  who  was  an 
interpreter  and  developer  of  ancient  thoughts,  and 
to  a  later  elaborator  of  the  Liber  Jordani  de  ratione 
pofideris,  whom  he  calls  the  ' '  forerunner  of  Leonardo 
da  Vinci,"  a  great  influence  on  Leonardo,  Cardano, 
and  Benedetti.  The  most  important  corrections  to 
Jordani  opiisculuni  de  ponderositate^  which  Tartaglia 
published  as  his  own  and  used  in  Questi  et  inventioni 


6  THE  SCIENCE  OF  MECHANICS 

diverse  without  naming  Jordanus  or  his  later  elabora- 
tor,  are  contained  in  a  manuscript  under  the  title 
Liber  Jordani  de  ratio7ie  pojideris^  which  Duhem 
found  in  the  national  library  at  Paris  {fond  latin ^ 
No.  7378  A).  This  leads  to  the  supposition  of  the 
anonymous  ''forerunner."  Also,  Leonardo's  manu- 
scripts, which  were  not  carefully  preserved  and  were 
unprotected  from  unauthorised  use,  have  had,  accord- 
ing to  Duhem,  in  spite  of  their  delayed  publication, 
an  effect  on  Cardano  and  Benedetti.  The  authors 
named  above  influenced,  above  all,  Galileo  in  Italy, 
Stevinus  in  Holland,  and  their  works  reached  France 
by  both  channels.  There  they  found,  in  the  first 
place,  fruitful  soil  in  Roberval  and  Descartes.  Con- 
sequently, the  continuity  between  ancient  and 
modern  statics  was  never  broken. 

Let  us  now  consider  some  details.  The  author 
of  the  Mechanical  Problems  mentioned  on  p.  511 
remarks  about  the  lever  that  the  weights  which  are 
in  equilibrium  are  inversely  proportional  to  the  arms 
of  the  lever  or  to  the  arcs  described  by  the  end- 
points  of  the  arms  when  a  motion  is  imparted  to 
them.^  With  great  freedom  of  interpretation  we 
can  regard  this  remark  as  the  incomplete  expression 

^  According  to  the  view  of  E.  Wohlwill,  it  may  be  considered  to  be 
decided  that  the  Mechatttcal  Problems  cannot  be  due  to  Aristotle.  Cf. 
Zeller,  Philosophic  der  Griechen,  3rd  ed,,  pt.  ii,  §  ii,  note  on  p.  90. 
But  then  a  thorough  investigation  as  to  whether  the  lately  found  Arabic 
translation  (published  in  1893)  of  Hero's  Mechanics,  it  not  the  older 
text,  is  necessary.  Cf.  Heron's  Werke,  edited  by  L.  Nix  and  W.  Schmidt 
(Leipsic,  1900),  vol.  ii. 


ADDITIONS  AND  ALTERATIONS        7 

of  the  principle  of  virtual  displacements.      But,  with 
Jordanus    Nemorarius    (Duhem,    op.    cit.,    pp.    121, 
122),   the  equilibrium  of   the  lever  is  characterised 
by  the  inverse  proportionality  of  the  height  to  which 
the  weights  are  raised  (or  the  depths  to  which  they 
fall)  to  the  weights  which  are  in  equilibrium.      The 
essential  point  is  brought  into  prominence  by  this. 
Jordanus  also  knows  that  a  weight  does  not  always 
act  in  the  same  way,  and  introduces — though  only 
qualitatively — the    conception    of   weight  according 
to  position:   "secundum  situm  gravius,   quando  in 
eodem^  situ  minus  obliquus  est  descensus  "  {op.  cit. , 
p.    118).      The  *' forerunner  "  of  Leonardo  improves 
and    completes    the    exposition    of    Jordanus.      He 
recognises  the  equilibrium  of  an  angular  lever  whose 
axis  lies  above  the  weights,  by  the  consideration  of 
the  possible  depths  of  falling  and  heights  of  rising, 
as    stable  (op.    cit.^   p.    142).       He   knows  also   that 
such    a    lever    directs  itself   in  such  a  manner  that 
the  weights  are  proportional  to  their  distances  from 
the  vertical  through  the  axis  {op.  cit.,  pp.   izj2,   143), 
and  thus  arrives  in  essentials  at  the  use  of  the  con- 
ception oi  moment.     The  **  gravitas  secundum  situm  " 
thus  here  attains  a  quantitative  form  and  is  used  in  a 
brilliant  way  for  the  solution  of  the  problem  of  the 
inclined  plane  {op.  cit.,  p.   145).      If  two  weights  on 
inclined  planes  of  equal  heights  but  different  lengths 
are  so  connected  by  a  rope  and  pulley  that  the  one 
must    rise    when   the  other  sinks,   the  weights  are, 


8  THE  SCIENCE  OF  MECHANICS 

in  the  case  of  equilibrium,  inversely  as  the  vertical 
displacements,  that  is  to  say,  vary  directly  as  the 
lengths  of  the  inclined  planes.  Consequently  in 
this  the  "forerunner"  anticipated  the  essential 
elements  of  modern  statics. 

The  study  of  the  manuscripts  of  Leonardo,  which 
have  only  been  published  in  part,  is  extremely  profit- 
able. The  comparison  of  his  various  occasional 
notes  shows  clearly  his  knowledge  of  the  principle 
of  virtual  displacements,  or  rather  of  the  concept 
of  work,  though  he  does  not  use  any  special  nomen- 
clature. "When  a  force  carries  (raises?)  a  body 
(a  weight  ?)  in  a  certain  time  through  a  definite  path, 
the  same  force  can  carry  (raise  ?)  half  of  the  body 
(the  weight  ?)  in  the  same  time  through  a  path 
double  in  length."  This  theorem  is  applied  to 
machines,  lever,  pulleys,  and  so  on,  and  by  this 
the  rather  doubtful  meaning  of  the  above  words 
is  more  closely  determined,  if  we  have  a  definite 
quantity  of  water  which  can  sink  to  a  definite  depth, 
we  can,  according  to  Leonardo,  drive  one  or  even 
two  equal  mills  with  it,  but  in  the  second  case  we 
can  only  accomplish  as  much  as  in  the  first  case. 
The  perception  of  the  "potential  lever,"  to  which 
Leonardo  attained  by  a  stroke  of  genius,  put  him 
in  the  position  to  gain  all  the  insight  which  was 
reached  later  by  the  conception  of  "moment."  His 
figures  make  us  suspect  that  the  consideration  of  the 
pulley  and  the  wheel  and  axle  showed  him  the  way 


ADDITIONS  AND  ALTERATIONS        g 

to  his  conception  (cf.  Mechanics,  p.  20).  Leonardo's 
constructions  concerning  the  pulls  on  combinations 
of  cords  visibly  rest,  too,  on  the  thought  of  the 
potential  lever.  Leonardo  was  less  happy  in  the 
treatment  of  the  problem  of  the  inclined  plane. 
By  the  side  of  sketches  in  which  sometimes  a  correct 
view  is  expressed,  w^e  find  many  incorrect  con- 
structions. However,  we  must  consider  Leonardo's 
scribblings  as  leaves  of  a  diary,  which  fix  the  most 
various  sudden  ideas  and  points  of  view  and  begin- 
nings of  investigations,  and  do  not  attempt  to  carry 
out  these  investigations  according  to  a  unitary 
principle.  To  explain  the  fact  that  Leonardo  was 
not  master  of  all  the  problems  which  had  been  com- 
pletely solved  in  the  thirteenth  century,  we  must 
remember  that  it  by  no  means  suffices,  as  we  must 
recognise  with  Duhem,  that  an  insight  should  be  once 
attained  and  made  known,  but  years  and  centuries 
are  often  necessary  for  this  insight  to  be  generally 
recognised  and  understood  (Duhem,  op.  cit.,  p.  182). 
The  idea  of  the  impossibility  of  perpetual  motion 
is  developed  with  Leonardo  to  great  clearness.  The 
consideration  about  the  mill  shows  this  :  '  *  No  impetus 
without  life  can  press  or  draw  a  body  without  accom- 
panying the  body  moved  ;  these  impetuses  can  be 
nothing  else  than  forces  or  gravity.  When  gravity 
presses  or  draws,  it  effects  motion  only  because  it 
strives  for  rest  ;  no  body  can,  by  its  motion  of  fall- 
ing, rise  to  the  height  from  which  it  fell  ;  its  motion 


10         THE  SCIENCE  OF  MECHANICS 

reaches  an  end"  {op.  cit.,  p.  53).  ''Force  is  a 
spiritual  and  invisible  power  which  is  impregnated 
in  bodies  by  motion  (here  we  certainly  have  to  think 
of  what  at  the  present  time  is  called  vis  viva)  ;  the 
greater  it  is  the  more  quickly  does  it  expend  itself" 
{op.  cit.,  p,  54).  Cardano  has  a  similar  view  in 
which  we  may  judge  an  influence  of  Leonardo  to  be 
probable  if  we  have  grounds  for  doubting  Cardano's 
independence  {op.  cit.^  pp.  40,  57,  58).  Also, 
Aristotle's  idea  that  only  the  circular  motion  of  the 
heavens  is  eternal  appears  again  with  Cardano. 
Duhem  considers  that  Cardano  is  not  a  common 
plagiarist.  He  used  indeed  without  acknowledg- 
ment the  works  of  his  predecessors,  especially  those 
of  Leonardo,  but  brought  these  works  into  a  better 
connection  and,  by  that,  improved  the  position  of 
the  sixteenth  century  {op.  cit. ,  pp.  42,  43).  Cardano 
does  not  overcome  the  problem  of  the  inclined  plane  ; 
his  opinion  is  that  the  weight  of  the  body  on  the 
inclined  plane  is  to  the  whole  weight  as  the  angle  of 
elevation  of  the  plane  is  to  a  right  angle.  Benedetti 
put  himself  in  opposition  to  all  his  predecessors, 
and  this  opposition  had  a  good  effect,  especially  in 
criticism  of  the  dynamical  doctrines  of  Aristotle. 
But  Benedetti  was  often  opposed  to  what  was  right. 
In  his  writings  occur  again  thoughts  of  Leonardo's, 
and  errors  of  Leonardo's  as  well. 

If  we  regard  the  discoveries  we  have  just  spoken 
of  as  sufficiently  known  and  accessible  to  the  sue- 


ADDITIONS  AND  ALTERATIONS       ii 

cessors  of  the  above  men,  there  remains  for  these 
successors — especially  for  Stevinus  and  Galileo — not 
very  much  more  to  do  in  statics.  Stevinus's  solution 
of  the  problem  of  the  inclined  plane  {cf.  Mechanics^ 
pp.  24-31)  is  indeed  quite  original,  but  the  *' fore- 
runner "  of  Leonardo  already  knew  the  result  of  the 
considerations  of  Stevinus  and  Galileo,  and  Galileo's 
considerations  join  on  to  those  of  Cardano.  From 
the  consideration  of  the  inclined  plane  Stevinus 
attained  to  the  composition  and  resolution  of  rect- 
angular components  according  to  the  principle  of 
the  parallelogram,  and  considered  this  principle  to 
be  generally  valid  without  being  able  to  prove  it. 
Roberval  filled  up  this  gap.  He  imagined  a  weight 
R  supported  by  pulleys  and  held  in  equilibrium  by 
a  cord  of  any  direction  loaded  with  counter-weights 
P  and  O.  If,  first,  we  consider  one  cord  as  a 
rod  which  can  rotate  about  the  pulley  and  apply 
Leonardo's  principle  of  the  potential  lever,  and  then 
proceed  in  a  similar  way  with  respect  to  the  other 
cord,  we  find  the  relations  of  R  to  P  and  Q  and  all 
the  theorems  which  hold  for  the  triangle  of  forces 
or  the  parallelogram  of  forces  {op.  at.,  esp.  p.  319). 
Descartes  finds  in  the  principle  of  virtual  displace- 
ments the  foundation  for  the  understanding  of  all 
machines.  He  sees  in  work,  the  product  of  weight 
and  distance  of  falling  (in  his  nomenclature, 
"force*'),  the  determining  circumstance  or  cause 
of  the    behaviour   of  machines,    the    Why   and   not 


12         THE  SCIENCE  OF  MECHANICS 

merely  the  How  of  the  event.  It  is  not  a  question 
of  the  velocity,  but  of  the  height  of  raising  and  the 
depth  of  falling.  *'  For  it  is  the  same  thing  to  raise 
a  hundred  pounds  two  feet  or  two  hundred  pounds 
one  foot "  {op.  cit. ,  p.  328  ;  cf.  p.  54  of  Mechanics 
on  Pascal's  statement).  Descartes  denies  the  un- 
mistakable influence  on  his  thoughts  of  all  his  pre- 
decessors from  Jordanus  to  Roberval  ;  and  yet  his 
developments  show  everywhere  important  progress, 
and  throughout  he  emphasises  essential  points  {pp. 
cit.,  pp.  327-352). 

With  respect  to  details  we  must  refer  to  Duhem's 
brilliant  book.  Here  I  will  only  give  expression  to 
my  somewhat  different  opinion  on  the  relation  of 
ancient  to  modern  natural  science.  Natural  science 
grows  in  two  ways.  In  the  first  place,  it  grows 
by  our  retaining  in  memory  the  observed  facts  or 
processes,  reproducing  them  in  our  presentation,  and 
trying  to  reconstruct  them  in  our  thoughts.  But, 
as  the  observations  are  continued,  these  attempts 
at  construction,  which  are  successively  or  simultane- 
ously taken  in  hand,  always  show  certain  defects  by 
which  the  agreement  of  these  constructions  both  with 
the  facts  and  with  one  another  is  disturbed.  Thus 
there  results  a  need  for  material  correction  and 
logical  harmonisation  of  the  constructions.  This  is 
the  second  process  which  builds  up  natural  science. 
If  everyone  had  only  himself  to  rely  on,  he 
would   have   to    begin   anew   with   his   observations 


ADDITIONS  AND  ALTERATIONS       13 

and  thoughts  alone,  and  consequently  could  not 
get  far.  This  holds  both  for  single  human  beings 
and  for  single  nations.  Thus  we  cannot  treasure 
highly  enough  the  heritage  which  our  immediate 
predecessors  in  civilisation — the  Greek  students  of 
nature,  astronomers  and  mathematicians — have  be- 
queathed to  us.  We  enter  on  investigation  under 
favourable  conditions,  since  we  are  in  possession 
of  an  image  of  the  world, — although  this  image  be 
insufficient — and  are,  above  all,  equipped  with  the 
logical  and  critical  education  of  the  Greek  mathema- 
ticians^ This  possession  makes  the  continuance  of 
the  work  easier  for  us.  But  we  must  consider  not 
only  our  scientific  heritage  but  also  material  civilisa- 
tion— in  our  special  case  the  machines  and  tools 
which  have  been  handed  down  to  us  as  well  as  the 
tradition  of  their  use.  We  can  easily  set  up  ob- 
servations on  this  material  heritage,  or  repeat  and 
extend  those  which  led  the  investigators  of  ancient 
times  to  their  science,  and  thus  for  the  first  time 
learn  really  to  understand  this  science.  It  appears 
to  me  that  this  material  heritage — continually  wak- 
ing up  anew,  as  it  does,  our  independent  activity — is 
too  little  esteemed  in  comparison  with  the  literary 
heritage.  For  can  we  suppose  that  the  paltry 
remarks  of  the  author  of  the  Mechanical  Problems 
about  the  lever,  and  even  the  far  more  exact  re- 
marks of  the  Alexandrian  mathematicians,  would 
not  have  continually  obtruded  themselves  upon  the 


14         THE  SCIENCE  OF  MECHANICS 

observing  men  who  were  busied  with  machines, 
even  if  these  remarks  were  not  preserved  in  writing  ? 
Does  not  this  hold  good,  say,  about  the  knowledge 
of  the  impossibility  of  perpetual  motion,  which  must 
present  itself  to  everybody  who  does  not  seek 
wonder  in  mechanics,  as  a  dreamer  after  the  fashion 
of  the  alchemists,  but  is  busied,  as  a  calm  investi- 
gator, in  practice  with  machines  ?  Even  when  such 
finds  are  transferred  to  those  who  come  after,  they 
must  be  gained  independently  by  these  followers. 
The  sole  advantage  a  follower  has  consists  in  the 
start  that  he  has  gained  by  a  quicker  passage  over 
the  same  course,  by  which  he  outstrips  his  prede- 
cessors. An  incomplete  knowledge  put  into  words 
forms  a  relatively  firm  prop  for  fleeting  thoughts, 
from  which  the  thoughts,  seeking  among  facts,  set 
out,  and  to  which,  modifying  it  by  criticism  and 
comparison,  they  continually  return.  Now,  whether 
these  props  are  made  stronger  by  newer  experience 
or  are  gradually  shifted,  or  are  even  at  last  recognised 
as  invalid,  they  have  helped  us  on.  But  if  the  pre- 
decessor becomes  a  great  authority,  and  if  even  his 
errors  are  prized  as  marks  of  deep  insight,  we  get 
a  state  of  things  which  can  only  act  in  a  hurtful 
way  on  the  followers  of  this  man.  Thus,  by  many 
passages  in  the  writings  of  E.  Wohlwill  and  P. 
Duhem,  it  seems  that  even  Galileo  was  sometimes 
hindered,  even  in  his  later  years,  by  the  traditional 
peripatetic   burden  from  perceiving  undisturbed  his 


ADDITIONS  AND  ALTERATIONS       15 

own  far  stronger    light.      In  our   estimation  of  the 
importance   of  an    investigator,   then,   it    is   only  a 
question  of  what  new  use  he  has  made  of  old  views 
and  under    what    opposition    of  his    contemporaries 
and  followers  his  own  views  have  come  to  be  held. 
From  this  point  of  view,   Duhem  seems  to  me  to 
go  rather  too  far  in  his  feeling  of  reverence  towards 
the  memory  of  Aristotle.      With  Aristotle  {De  coelo, 
book  iii,  2)  there  are,  for  example,   among  unclear 
and  unpromising  utterances,  the  passages:   "What- 
ever the   moving   force   may   be,    the  less   and   the 
lighterYeceive  more  motion  from  the  same  force.  .  .  . 
The  velocity  of  the  less  heavy  body  will  be  to  that 
of   the  heavier    body  as  the  heavier  to  the  lighter 
body."       If  we    disregard    the    fact    that  Aristotle 
cannot  be  credited  with  a  clear  distinction  of  path, 
velocity,  and  acceleration,  we  can  recognise  in  this 
the  expression  of  a  primitive  but  correct  experience 
which  led  at  length  to  the  conception  of  mass.      But, 
after  what  we  have  said  in  the  whole  of  the  second 
chapter,    it    seems    hardly    thinkable    to    refer    this 
passage  to  the  raising  of  weights  by  machines,  to 
combine  it  with  what  Aristotle   has  said  about  the 
lever,  and  then  to  see  in  it  the  germ  of  the  concep- 
tion of  work  (Duhem,  op.  cit. ,  pp.  6,  y  ;  cf.  Vailati, 
Bolletino  di  bibliografia   e  storia   di   scienze    mate- 
maticJie,    Feb.   and    March,  1906,   p.     3).       Further, 
Duhem  blamesStevinus  for  his  peripatetic  tendencies. 
But  Stevinus  seems  to  me  to  be  in  the  right  when 


i6         THE  SCIENCE  OF  MECHANICS 

he  puts  himself  in  opposition  to  the  "wonderful" 
circles  of  Aristotle,  which  are  not  described  in  the 
case  of  equilibrium.  This  is  just  as  justifiable  as 
the  protest  of  Gilbert  and  Galileo  against  the 
hypothesis  of  the  effectiveness  of  a  mere  position 
or  a  point  (see  Mechanics,  p.  533).  Only  from  a 
broader  point  of  view,  when  work  is  recognised  as 
that  which  determines  motion,  does  the  dynamical 
derivation  of  equilibrium  attain  the  merit  of  greater 
rationality  and^enerality.  Before  that,  hardly  any- 
thing could  be  urged  against  Stevinus's  inspired 
deductions  on  the  grounds  of  instinctive  experience 
and  after  the  manner  of  Archimedes. 

V 

[To  p.   112,  last  line  of  paragraph  i,  add  :] 

To  form  some  idea  of  the  slowness  with  which 
the  new  notions  about  air  became  more  familiar  to 
men,  it  is  enough  to  read  the  article  on  air  which 
Voltaire,^  one  of  the  most  enlightened  men  of  his 

^  [Voltaire's  article  ' '  Air  "  in  the  first  volume  of  his  Questions  sur 
r Encyclopedic  par  des  Amateurs  was  republished  in  the  Collection 
complette  des  CEuvres  de  Mr  de  .  .  .  (vol.  xxi,  Geneva,  1774,  PR-  73" 
81  ;  the  part  noticed  in  the  text  above,  which  contains  Voltaire's  own 
opinions,  is  on  pp.  77-79).  The  Questions  were  first  published  in 
1770-72  in  seven  volumes,  and  the  article  "Air  "is  in  the  first  part 
(1770).  The  Dictionnaire  Philosophique  was  first  published  in  1764, 
and  was  greatly  augmented  in  various  subsequent  editions  from  1767  to 
1776.  The  editor,  de  Kehl,  in  1785-89,  included  various  works  under 
the  single  title  of  Dictionnaire  Philosophique,  viz.,  the  Dictionnaire 
Philosophique,  the  Questions,  a  manuscript  dictionary  entitled  V  Opinion 
par  t  Alphabet,  Voltaire's  articles  in  the  great  Encyclopddie,  and  several 
articles  destined  for  the  Dictionnaire  de  PAcacUmie  Fran^aise.  The 
article  "Air"  is  contained  in  vol.  xxvi  of  M.  Beuchot's  CEuvres  de 
Voltaire  (72  volumes,  Paris,  1829),  pp.  136-147.] 


ADDITIONS  AND  ALTERATIONS       17 

time,  wrote  in  his  Dictionyiau'e  PJiilosophiqiie  from 
the  Ency  dope  die,  in  1764 — a  century  after  Guericke, 
Boyle,  and  Pascal,  and  not  long  before  the  dis- 
coveries of  Cavendish,  Priestley,  Volta,  and  Lavoisier, 
— that  air  is  not  visible  and,  quite  generally,  is  not 
perceptible  ;  all  the  functions  that  we  ascribe  to  the 
air  can  be  discharged  by  the  perceptible  exhalations 
whose  existence  we  have  no  grounds  for  doubting. 
How  can  the  air  enable  us  to  hear  the  different  notes 
of  a  melody  simultaneously  ?  Air  and  aether  are, 
with  respect  to  the  certainty  of  their  existence,  put 
on  the  same  level. 

VI 

On  p.  128  of  the  Mechanics,  the  words  "  Dynamics 
was  founded  by  Galileo,"  and  ''  Only  by  traces,  which 
were  for  the  most  part  mistaken,  do  we  find  that 
their  thought  extended  to  dynamics,"  and  on 
pp.  128-129,  the  words  "and  that  .  .  .  inquiry" 
are  omitted. 

VII 
[To  p.   129,  line  2,  add  :] 

Besides,  the  views  of  Aristotle  found  opponents 
even  in  antiquity.  Especially  the  Aristotelian 
opinion  that  the  continued  motion  of  a  body  which 
is  projected  is  brought  about  by  means  of  the  air 
which  has  been  set  in  motion  at  the  same  time 
plainly  showed  an  obvious  point  of  attack  to  criticism. 
According    to  Wohlwill's  researches,   Philoponos,   a 


1 8        THE  SCIENCE  OF  MECHANICS 

writer  of  the  sixth  century  of  the  Christian  era, 
expressly  contested  this  view — a  view  contrary  to 
every  sound  instinct.  Why  must  the  moving  hand 
touch  the  stone  at  all  if  the  air  manages  everything  ? 
This  natural  question  asked  by  Philoponos  did  not 
fail  to  exercise  an  influence  on  Leonardo,  Cardano, 
Benedetti,  Giordano  Bruno,  and  Galileo.  Philoponos 
also  contradicts  the  assertion  that  bodies  of  greater 
weight  fall  more  quickly,  and  refers  to  observation. 
Finally,  Philoponos  shows  a  modern  trait  in  that  he 
denies  any  force  to  \}ci^ position  in  itself^  but  attributes 
to  bodies  the  effort  to  preserve  their  order  {cf. 
Wohlwill,  "  Ein  Vorganger  Galilei's  im  6.  Jahr- 
hundert,"  PJiysik.  Zeitschrift  von  Riecke  und  Sinion^ 
7.  Jahrg,  No.   i,  pp.  23-32). 

VIII 

[After  "  gravity  "  on  line  i  of  p.  521,  insert  passage, 
which  \s  partly  given  on  p.  521  :] 

Just  so  is  the  increasing  of  the  projectile-force  of 
a  stone  by  the  thrower  reduced  to  an  aggregation 
of  impulses.  Such  an  impulse  has,  according  to 
Benedetti,  the  tendency  to  force  the  body  forward 
in  a  straight  line.  A  body  projected  horizontally 
approaches  the  earth  more  slowly  ;  consequently,  the 
gravity  of  the  earth  appears  to  be  partly  taken  away. 
A  spinning  top  does  not  fall,  but  stands  on  the  end 
of  its  axis,  because  its  parts  have  the  tendency  to  fly 


ADDITIONS  AND  ALTERATIONS       19 

away  tangentially  and  perpendicularly  to  the  axis, 
and  by  no  means  to  approach  the  earth.  Benedetti 
ascribes  the  continued  motion  of  a  projected  body 
not  to  the  influence  of  the  air  but  to  a  ''virtus 
impressa,"  but  does  not  attain  to  full  clearness  with 
respect  to  the  problems  (G.  Benedetti,  Sulle  pro- 
per zioiii  dei  motu  locali,  Venice,  1553;  Divers, 
speculai.   math,   et  physic,   liber,  Turin,    1585). 

Galileo,  in  the  works  of  his  youth,  which  was 
spent  in  Pisa,  appears,  as  has  become  known  by  the 
recent  critical  edition  of  his  works,  as  an  opponent 
of  Aristotle,  as  doing  honour  to  the  *'  divine  "  Archi- 
medes, and  as  the  immediate  follower  of  Benedetti, 
whom  he  follows  both  in  the  manner  in  which  he 
puts  questions  to  himself  and  often  in  the  way  of 
writing,  without,  however,  citing  him.  Like  Bene- 
detti, he  supposes  a  gradually  decreasing  "vis 
impressa  "  in  cases  of  projection.  If  the  projection 
is  upwards,  the  impressed  force  is  a  transferred 
"  lightness  "  ;  as  this  lightness  decreases,  the  gravity 
receives  an  increasing  preponderance  directed  below, 
and  the  motion  of  falling  is  accelerated.  In  this 
idea  Galileo  encounters  the  ancient  astronomer 
Hipparchus  of  the  second  century  B.C.,  but  does  not 
do  justice  to  Benedetti's  view  of  the  acceleration  of 
falling.  For,  according  to  Hipparchus  and  Galileo, 
the  motion  of  falling  would  have  to  be  uniform  when 
the  impressed  force  is  wholly  overcome. 


20         THE  SCIENCE  OF  MECHANICS 

IX 

[Note  to  p.   129,  line  2  up.      From  this  to  p.   130, 
line  10,  is  omitted,  and  the  passage  added  :] 

In  the  former  editions  of  this  book,  the  exposition 
of  Galileo's  researches  was  based  on  his  final  work, 
Discorsi  e  dimostrazioni  matematicJie  of  1 638.  ^  How- 
ever, his  original  notes,  which  have  become  known 
later,  lead  to  different  views  on  his  path  of  develop- 
ment. With  respect  to  these  I  adopt,  in  essentials, 
the  conclusions  of  E.  Wohlwill  {Galilei  und  sein 
Kmnpf  fiir  die  Kopernikanische  Lehre,  Hamburg 
and  Leipsic,  1909).  In  the  riper  and  more  fruitful 
time  of  his  residence  in  Padua,  Galileo  dropped  the 
question  as  to  the  "why"  and  inquired  the  "how" 
of  the  many  motions  which  can  be  observed.  The 
consideration  of  the  line  of  projection  and  its  con- 
ception as  a  combination  of  a  uniform  horizontal 
motion  and  an  accelerated  motion  of  falling  enabled 
him  to  recognise  this  line  as  a  parabola,  and  conse- 
quently the  space  fallen  through  as  proportional  to 
the  square  of  the  time  of  falling.  The  statical  in- 
vestigations on  the  inclined  plane  led  to  the  con- 
sideration of  falling  down  such  a  plane,  and  also  to 
the  observation  of  the  vibrating  pendulum.  From 
comprehensive  observations  and  experiments  on  the 

^  [There  is  a  convenient  German  annotated  translation  of  the  Discorsi 
e  dimostrazioni  tnatetnatiche  by  A.  J.  von  Oettingen  in  Ostwald^s  Klassi- 
ker  der  exakten  Wissenschaften,  Nos,  1 1,  24,  25  ;  and  an  English  trans- 
lation by  Henry  Crew  and  Alfonso  de  Salvio  under  the  title  Dialogues 
concerning  Two  New  Sciences^  New  York,  1914.] 


ADDITIONS  AND  ALTERATIONS       21 

pendulum  it  appeared  that  a  body  which  falls  down 
a  series  of  inclined  planes  can,  by  means  of  the 
velocity  thus  obtained,  rise  on  any  series  of  other 
planes  to  the  original  height  and  no  higher.  In 
other  words,  the  velocity  obtained  by  the  falling 
only  depends  on  the  distance  fallen  through. 
Finally,  Galileo  reached  a  definition  of  uniformly 
accelerated  motion  which  has  the  properties  of  the 
motion  of  falling,  and  from  which,  inversely,  all 
those  provisional  lemmas  which  led  him  to  his  view 
can  be  deductively  derived. 

With  respect  to  the  definition  of  uniformly  acceler- 
ated motion,  GaHleo  hesitated  for  a  long  time.  He 
first  called  that  motion  uniformly  accelerated  in  which 
the  increments  of  velocity  are  proportional  to  the 
lengths  of  path  described  ;  he  held,  according  to  a 
fragment  dating  from  1604  {Edizione  Nazionale^ 
vol.  viii,  pp.  373-374),  and  a  letter  to  Sarpi  written  at 
the  same  time,  that  this  conception  corresponded  to  all 
facts,  in  which,  however,  he  was  mistaken.  Accord- 
ing to  Wohlwill,  it  was  probably  about  1609  that  he 
overcame  the  error  and  defined  uniformly  accelerated 
motion  by  the  proportionality  of  the  velocity  to  the 
time  of  motion.  He  then  turned  away  from  his 
first  view  on  grounds  just  as  insufificient  as  those  on 
which  he  had  accepted  it  earlier.  The  natural  ex- 
planation of  all  this  will,  as  in  the  older  editions  of 
this  book,  be  spoken  of  later.  We  will  now  con- 
sider what  heritage  Galileo  left  to  modern  thinkers. 


22         THE  SCIENCE  OF  MECHANICS 

Here  it  will  appear  clearly  that  he  allowed  himself 
to  be  led  by  suppositions  which  to-day  can  be  con- 
ceived as  more  or  less  immediate  corollaries  from 
his  law  of  falling  ;  and  this  perhaps  speaks  most 
eloquently  for  his  talent  as  an  investigator  and 
for  his  discoverer's  instinct.  Now,  whether  Galileo 
attained  to  knowledge  of  the  uniformly  accelerated 
motion  of  falling  by  consideration  of  the  parabola  of 
projection  or  in  another  way,  we  cannot  doubt  that 
he  tested  the  law  of  falling  experimentally  as  well, 
Salviati,  who  represents  Galileo's  doctrines  in  the 
Discorsi,  assures  us  of  his  repeatedly  taking  part  in 
experiments,  and  describes  the  experiments  very 
accurately  {Le  opere  di  Galilei,  Edizione  Nazionale, 
vol.  viii,  pp.  212-213). 

X 

[To  p.  527,  line  25,  add  :] 

If,  now,  we  ask  what  views  into  the  nature  of 
things  Galileo  has  bequeathed  to  us,  or  at  least 
facilitated  in  a  lasting  manner  by  classically  simple 
examples,  we  find  : 

(i)  The  emphasis  upon  the  conception  of  work  in 
a  statical  connection.  There  is  no  saving  work  with 
machines  ; 

(2)  The  advancement  of  the  conception  of  work  in 
a  dynamical  connection.  The  velocity  attained  by 
falling,  when  resistance  is  neglected,  only  depends 
on  the  distance  fallen  through  ; 


■  ADDITIONS  AND  ALTERATIONS      23 

(3)  The  law  of  inertia  ; 

(4)  The  principle  of  the  superposition  of  motions. 
Galileo's  creative  activity  extends  far  beyond  the 

limits  of  mechanics  ;  we  will  only  call  to  mind  his 
founding  of  thermometry,  his  sketch  of  a  method 
for  the  determination  of  the  velocity  of  light/  his 
direct  proof  of  the  numerical  ratio  of  the  vibrations 
of  the  musical  interval  and  his  explanation  of  syn- 
chronous vibrations.  He  heard  of  the  telescope,  and 
that  was  enough  for  him  to  rediscover  and  to  im- 
provise one  with  two  lenses  and  an  organ-pipe.  In 
quick  succession  he  discovered,  by  the  help  of  his 
instrument,  the  mountains  of  the  moon — whose 
height  he  micasured, — Jupiter  with  his  satellites — a 
small  model  of  the  solar  system, — the  peculiar  form 
of  Saturn,  the  phases  of  Venus,  and  the  spots  and 
rotation  of  the  sun.  These  were  new  and  very 
strong  arguments  for  Copernicus.  Also  his  thoughts 
on  geometrically  similar  animals  and  machines  and  on 
the  form  and  firmness  of  bones  must  be  considered 
to  be  stimuli  to  the  development  of  new  mathe- 
matical methods.  Besides  Wohlwill,  E.  Goldbeck 
(•*  Galilei's  Atomistik  und  ihre  Ouellen,"  Biblioth. 
Math.,  3rd  series,  vol.  iii,  1902,  part  i)  has  recently 
shown  that  this  revolutionising  thinker  was  not 
wholly  independent  of  ancient  and  mediaeval  in- 
fluences.     In  particular,  the  first  day  of  the  Discorsi 

^  [See  Mach's   Popular  Scientific   Lectttres,  3rd   ed.,  Chicago   and 
London,  1S98.  pp.  50-54.] 


24        THE  SCIENCE  OF  MECHANICS 

contains  a  lengthy  exposition  of  Galileo's  atomistic 
reflections  which  clearly  stand  in  opposition  to 
Aristotle,  and  as  clearly  approximate  to  Hero's 
position.  These  reflections  led  him  to  extraordinary 
discussions  on  the  continuum  and  to  speculations,  in 
which  mysticism  and  mathematics  were  combined, 
on  the  finite  and  the  infinite,  which  remind  us, 
on  the  one  hand,  of  Nicolas  of  Cusa,  and,  on  the 
other  hand,  of  many  modern  mathematical  researches 
which  are  hardly  free  from  mysticism.^  That  Galileo 
could  not  attain  complete  clearness  in  all  his  thoughts 
need  surprise  us  no  more  than  his  occupation  with 
paradoxes,  whose  disturbing  and  clarifying  force 
every  thinker  must  have  experienced. 

With  respect  to  the  knowledge  of  accelerated 
motion  Galileo  has  done  the  greatest  service.  For 
the  sake  of  completeness  we  will  refer  to  P.  Duhem's 
researches  (''De  I'acceleration  produite  par  une 
force  constante  ;  notes  pour  servir  a  I'hi^toire  de  la 
dynamique,"  Congres  international  de  philosophies 
Geneva,  1905,  p.  859).  Without  entering  into  the 
many  historically  interesting  details  communicated 
by  Duhem,  we  will  here  only  add  the  following. 
According  to  the  literal  Aristotelian  doctrine,  a  con- 

^  [See  the  German  translation  of  the  first  two  days  of  the  Discorsi  in 
Ostivald's  Klassiker,  No.  I  r  (the  other  days  are  translated  in  Nos. 
24  and  25), 'especially  pp.  30-32.  Besides  the  article  of  Goldbeck 
mentioned  in  the  text  above,  there  is  an  article  by  E.  Kasner  on 
"  Galileo  and  the  Modern  Concept  of  Infinity,"  which  is  noticed  in  the 
Jahrbuch  iiber  die  Fortschritte  der  Mathe??iatik,  vol.  xxxvi,  1905, 
p.  49.  See  also  Crew  and  de  Salvio's  translation  of  -  the  Discorsi, 
pp.  26-40.] 


ADDITIONS  AND  ALTERATIONS       25 

stant  force  conditions  a  constant  velocity.     But  since 
the  increasing  velocity  of  falling  can  hardly  escape 
even  rough  observations,  the  difficulty  arises  of  bring- 
ing this  acceleration  into  harmony  with  the  doctrine 
that  held   the  field.      On  approaching  the   ground, 
the    body,    in    the    opinion    of   Aristotle,    becomes 
heavier.      The  traveller  hastens   when  approaching 
his  destination,  as  Tartaglia  expresses  it.      The  air 
which  at  one  time  was  viewed  as  a  hindrance  and  at 
another  time  as  a  motive  power  must,   in  order  to 
make  the  contradictions  more  supportable,  play  at 
one  time  the  one  part  and  at  another  time  the  other. 
The  hindering  space  of  air  between  the  body  and  the 
ground  is,  according  to  the  commentator  Simplicius, 
greater  at  the  beginning  of  the  motion  of  falling  than 
at   the  end  of  this   motion.      The   "forerunner"  of 
Leonardo  found  that  air  which  has  once  been  set  in 
motion  is  less  of  a  hindrance  for  the  body  moved. 
The  naif  observer  of  a  stone  projected  obliquely  or 
horizontally  and  describing  an  initial  line  which  is 
almost  straight  must  receive  the  natural  impression 
that  gravity  is  removed  by  the  impulse  to  motion 
(see  above.  Appendix  VIII).     Hence  the  distinction 
between  natural  and  forced  motion.     The  considera- 
tions of  Leonardo,  Tartaglia,  Cardano,  Galileo,  and 
Torricelli  on  projectiles  showed  how  the  idea  of  an 
alteration  of  two  motions  which  were  considered  to 
be  fundamentally  different  gradually  yields  to  that 
of  a  mixture  and  simultaneity  of  them.      Leonardo 


26         THE  SCIENCE  OF  MECHANICS 

was  acquainted  with  the  accelerated  motion  of  fall- 
ing, and  conjectured  the  increase  of  velocity  propor- 
tionally to  the  time, — which  he  ascribed  to  the 
successively  diminished  resistance  of  the  air, — but 
did  not  know  how  to  determine  the  correct  depend- 
ence of  the  space  fallen  through  on  the  time.  It  was 
first  at  about  the  middle  of  the  sixteenth  century 
that  the  thought  appeared  that  gravity  continually 
communicates  impulses  to  the  falling  body,  and  these 
impulses  are  added  to  the  impressed  force  which  is 
already  present  and  which  gradually  decreases.  This 
view  was  embraced  by  A.  Piccolomini,  J.  C.  Scaliger, 
and  G.  Benedetti.  Already  Leonardo  remarked, 
quite  by  the  way,  that  the  arrow  is  not  projected 
only  at  the  greatest  tension  of  the  bow,  but  also  in 
the  other  positions  by  the  touching  string  (Duhem, 
loc.  cit.,  p.  882).  But  it  was  only  when  Galileo 
gave  up  this  supposition  of  a  gradual  and  spontaneous 
decrease  of  the  impressed  force  and  reduced  this 
decrease  to  resisting  forces,  and  investigated  the 
motion  of  falling  experimentally  and  without  taking 
its  causes  into  consideration,  could  the  laws  of  the 
uniformly  accelerated  motion  of  falling  appear  in  a 
purely  quantitative  form. 

Further,  from  Duhem's  historical  exposition 
results  the  fact  that  Descartes  rendered,  inde- 
pendently of  Galileo,  more  important  services  in  the 
development  of  modern  dynamics  than  is  usually 
supposed,  and  than  I  too  have  supposed  in  the  third 


ADDITIOKTS  AND  ALTERATIONS      27 

chapter  of  my  Mechanics.  1  am  very  grateful  for 
this  instruction.  Descartes  busied  himself  during 
his  residence  in  Holland  (16 17-19),  in  co-operation 
with  Beeckmann  and  in  connection  with  the  re- 
searches of  Cardano  and  probably  also  of  Scaliger 
and  Benedetti,  with  the  acceleration  of  falling  bodies. 
He  thoroughly  recognised  the  law  of  inertia,  as 
results  from  letters  written  to  Mersenne  in  1629, 
before  Galileo's  publication  (E.  Wohlwill,  in  Die 
Entdeckimg  des  Beharrungsge seizes ,  pp.  142,  143, 
considered  it  possible  that  Galileo  indirectly  stimu- 
lated him).  Descartes  also  recognised  the  law  of 
uniformly  accelerated  motion  under  the  influence  of 
a  constant  force,  and  was  only  mistaken  with  respect 
to  the  law  of  dependence  of  the  path  described  on 
the  time. 

The  thoughts  of  Galileo  and  Descartes  mutually 
complete  each  other.  Galileo  investigated  the 
motion  of  descent  phenomenologically,  and  without 
inquiring  into  its  causes,  while  Descartes  derived 
this  motion  from  the  constant  force.  Naturally  in 
both  investigations  a  constructive  and  speculative 
element  was  active,  but  this  element  with  Galileo 
kept  close  to  the  concrete  case,  while  with  Descartes 
it  came  in  earlier  with  more  general  experiences. 
Certainly  Descartes,  in  his  Pn?iciples  of  Philosophy ^^ 

^  [This  work  was  first  published  at  Amsterdam  in  1644  under  the 
title  :  Renati  DesCartes  Principia  Phiioso/huv,  and  this  was  the  only 
edition  that  appeared  in  Descartes'  lifetime.  A  translation  into  French 
was  made  by  one  of  Descartes'  friends,  the  Abbe  Claude  Picot.    Descartes 


28         THE  SCIENCE  OF  MECHANICS 

observed  the  transference  of  motion  and  the  loss  of 
motion  of  the  impinging  body  and  the  general  philo- 
sophical consequences  that  (i)  without  the  giving  of 
motion  to  other  bodies  there  can  be  no  loss  of  motion 
(inertia)  ;  (2)  every  motion  is  either  original  or  trans- 
ferred from  somewhere  ;  (3)  the  original  quantity  of 
motion  cannot  be  increased  or  diminished.  From  this 
standpoint  he  could  imagine  that  every  apparently 
spontaneous  motion  whose  origin  was  not  perceptible 
was  introduced  by  invisible  impacts. 

The  great  advantage  which  I — perhaps  in  opposi- 
tion to  Duhem — ascribe  to  the  method  of  Galileo 
consists  in  the  careful  and  complete  exposition  of 
the  mere  facts.  In  this  exposition  nothing  remains 
concealed  behind  the  expression  ' '  force "  which 
could  be  conjectured  or  disentangled  by  speculation. 
On  this  point  opinions  are  divided  even  at  the 
present  time. 

XI 

[On  p.   194,  line  10,  add  :] 

Baliani,  in  his  preface  to  De  inotu  gravium  of 
1638,  distinguished,  according  to  G.  Vailati,  between 
the  weight  as  agens  and  the  weight  d.s  pattens,  and 
is  therefore  a  forerunner  of  Newton. 

read  this  translation  and  found  it  much  to  his  taste,  and,  when  it  was 
completed  in  1674,  wrote  a  preface  to  it.  This  French  translation 
passed  through  many  editions  ;  the  fourth  was  published  at  Paris  in 
1681,  and  bears  the  title:  Les  Principes  de  la  Philosophie  de  Reni 
Descartes.  Qtiatrieme  edition,  Reveue  et  corrig^e  fort  exaclefnent  par 
Monsieur  CLR.'\ 


ADDITIONS  AND  ALTERATIONS      29 

XII 

[To  p.  201,  line  18,  add  :] 

Newton's  achievements  are  not  limited  to  the 
domain  which  is  the  subject  of  this  book.  Even  his 
Principia  treats  questions  which  do  not  belong  to 
mechanics  proper.  Motion  in  resisting  media  and 
the  motion  of  fluids — even  under  the  influence  of 
friction — are  treated  there,  and  the  velocity  of  the 
propagation  of  sound  is  theoretically  deduced  for 
the  first  time.  The  optical  works  of  Newton  contain 
a  series  of  the  most  important  discoveries.  He 
demonstrated  the  prismatic  decomposition  of  light 
and  the  compounding  of  white  light  from  rays  of 
light  of  different  colours  and  unequal  refrangibilities, 
and,  in  this  connection,  gave  a  proof  of  the  periodicity 
of  light  and  determined  the  length  of  period  as  a 
function  of  the  colour  and  refrangibility.  Also  it 
was  Newton  who  first  grasped  the  essential  point  in 
the  polarisation  of  light.  Other  studies  led  him  to 
establish  his  law  of  cooling  and  the  thermometric  or 
pyrometric  principle  founded  on  this  law.^  In  his 
papers    and    book    on    optics  ^   Newton   showed  the 

^  [C/".  Mach,  Die  Principien  der  Wdrmelehre^  2nd  ed.,  Leipsic,  1900, 
pp.  58-61.]  ^ 

2  [Newton's  Opticks :  or  a  Treatise  of  the  Reflexions,  Refractions^ 
Inflexions,  and  Colours  of  Light ;  also  Treatises  of  the  Species  and 
Magnitude  of  Cur-ci linear  Figures  was  published  at  London  in  1 704, 
and  again,  with  additions  but  without  the  mathematical  appendices,  in 
1717,  1718,  1721,  and  1730.  A  Latin  translation,  by  Samuel  Clarke, 
was  first  published  at  London  in  1706  ;  and  a  useful  annotated  German 
translation   by  W.  Abendroth    was    published   as   Nos.  96  and   97  of 


30         THE  SCIENCE  OF  MECHANICS 

paths  which  led  to  his  discoveries  quite  frankly  and 
without  any  restraint.  Apparently  the  unpleasant 
controversies  in  which  these  first  publications  of  his 
involved  him  had  an  influence  on  his  exposition  in 
the  Principia.  In  the  Principia  he  gave  the  proofs 
of  the  theorems  that  he  had  discovered  in  a  synthetic 
form,  and  did  not  disclose  the  methods  which  had  led 
him  to  these  theorems.  The  acrimonious  controversy 
between  Newton  and  Leibniz,  and  between  their 
respective  followers,  on  the  priority  of  the  discovery 
of  the  infinitesimal  calculus,  was  chiefly  caused  by 
the  late  publication  of  Newton's  method  of  fluxions. 
To-day  it  is  quite  clear  that  both  Newton  and  Leibniz 
were  stimulated  by  their  predecessors  and  had  no 
need  to  borrow  from  one  another,  and  also  that  the 
discoveries  were  sufficiently  prepared  for  to  enable 
them  to  appear  in  different  forms.  The  preparatory 
works  of  Kepler,  Galileo,  Descartes,  Fermat,  Rober- 
val,  Cavalieri,  Guldin,  Wallis,  and  Barrow  were 
accessible  to  both  Newton  and  Leibniz.^ 

OstwalcTs  Klassiker  der  exakten  Wissenschaften  in  1898.  Newton's 
Optical  Lectures  read  in  the  Ptiblick  Schools  of  the  University  of 
Cambridge  Anno  Domini,  i66g,  was  translated  into  English  from 
the  original  Latin  and  published  at  London  in  1728,  after  Newton's 
death.  The  Latin  was  published  at  London  in  1729.  Newton's  papers 
on  optics  are  printed  in  vols,  vi-xi  of  the  Philosophical  Transactions^ 
and  begin  in  the  year  1672.] 

^  [On  Newton's  mathematical  and  physical  achievements,  we  may 
refer  to  M.  Cantor's  Vorlesiingen  iiber  Geschichte  der  Mathejuatik, 
vol.  iii,  2nd  ed.,  Leipsic,  1901,  pp.  156-328,  and  F.  Rosenberger's 
excellent  compilation,  Isaac  Newton  tind seine physikalischen  Principien, 
Leipsic,  1895.] 


ADDITIONS  AND  ALTERATIONS      31 

XIII 

[On  p.  539,  end  of  Appendix  XVII,  add  :] 

It  is  remarkable  that  Galileo,  in  his  theory  of 
the  tides,  treated  the  first  dynamical  problem  about 
the  world  without  troubling  about  the  new  system 
of  co-ordinates.  He  considered  in  the  most  naive 
manner  the  fixed  stars  as  the  new  system  of  reference. 

XIV 

[To  p.  .222,  line  2,  add  :] 

These  sentences  were  contained  in  the  first  edition 
of  1883,  and  thus  long  before  the  discussion  of 
electro-magnetic  mass  had  begun. 

I  may  here  refer  to  A.  Lampa's  paper  '*Eine 
Ableitung  des  Massenbegriffs  "  in  the  Prague  Journal 
Lotos,  191 1,  p.  303,  and  especially  to  the  excellent 
remarks  on  the  general  method  of  treatment  of  such 
questions  on  pp.  306  et  seqq» 

XV 

[On  p.  225,  instead  of  note,  put:] 

On  the  physiological  nature  of  the  sensations  of 
time  and  space  cf.  Analyse  der  Emfindungen^  6th 
ed.  ;  ^  Erkenntnis  und  Irrtum,  2nd  ed. 

^  [An  English  translation,  published  by  the  publishers  of  the  present 
volume,  of  this  edition  under  the  title  :  The  Analysis  of  Sensations, 
.  .  .   in  1914.] 


32         THE  SCIENCE  OF  MECHANICS 

XVI 

On  p.  542,  end  of  Appendix  XIX,  omit  the 
words  "to  which  1  shall  reply  in  another  place," 
and  add  the  reference  :  Erkenntnis  und  Irrtuni^  2nd 
ed. ,  Leipsic,  1906,  pp.  434-448. 

XVII 

[To  p.  229,  line  2,  add  :] 

If,  in  a  material  spatial  system,  there  are  masses 
with  different  velocities,  which  can  enter  into  mutual 
relations  with  one  another,  these  masses  present  to 
us    forces.      We    can    only   decide  how  great  these 
forces    are  when    we  know  the  velocities  to  which 
those   masses  are   to   be   brought.      Resting  masses 
too  are  forces  if  all  the  masses  do  not  rest.      Think, 
for  example,  of  Newton's  rotating  bucket  in  which 
the  water  is  not  yet  rotating.      If  the  mass  ni  has 
the  velocity  v^  and  it  is  to  be  brought  to  the  velocity 
z/2,    the    force    which    is   to   be   spent   on   it   is  /  = 
m{v^—V2)lt,   or  the  work  which  is  to  be  expended 
is  ps  —  m{v^ —  v^).      All  masses  and  all  velocities, 
and  consequently  all  forces,  are  relative.      There  is 
no  decision  about  relative  and  absolute    which  we 
can  possibly  meet,  to  which  we  are  forced,  or  from 
which    we   can    obtain    any    intellectual    or    other 
advantage.      When  quite  modern  authors  let  them- 
selves  be  led  astray  by  the  Newtonian  arguments 
which    are    derived   from   the   bucket    of   water,    to 


ADDITIONS  AND  ALTERATIONS      33 

distinguish  between  relative  and  absolute  motion, 
they  do  not  reflect  that  the  system  of  the  world  is 
only  given  once  to  us,  and  the  Ptolemaic  or  Coper- 
nician  view  is  our  interpretation,  but  both  are 
equally  actual.  Try  to  fix  Newton's  bucket  and 
rotate  the  heaven  of  fixed  stars  and  then  prove  the 
absence  of  centrifugal  forces. 

XVIII 
[To  p.  229,  line  18,  add:] 

We  rrtust  suppose  that  the  change  in  the  point  of 
view  from  which  the  system  of  the  world  is  regarded 
which  was  initiated  by  Copernicus,  left  deep  traces 
in  the  thought  of  Galileo  and  Newton.  But  while 
Galileo,  in  his  theory  of  the  tides,  quite  naively 
chose  the  sphere  of  the  fixed  stars  as  the  basis  of 
a  new  system  of  co-ordinates,  we  see  doubts  ex- 
pressed by  Newton  as  to  whether  a  given  fixed  star 
is  at  rest  only  apparently  or  really  {Principia,  1687, 
p.  II).  This  appeared  to  him  to  cause  the  diffi- 
culty of  distinguishing  between  true  (absolute)  and 
apparent  (relative)  motion.  By  this  he  was  also 
impelled  to  set  up  the  conception  of  absolute  space. 
By  further  investigations  in  this  direction  —  the 
discussion  of  the  experiment  of  the  rotating  spheres 
which  are  connected  together  by  a  cord  and  that  of 
the  rotating  water-bucket  (pp.  9,  11) — he  believed 
that  he  could  prove    an   absolute  rotation,   though 

3 


34        THE  SCIENCE  OF  MECHANICS 

he  could  not  prove  any  absolute  translation.  By 
absolute  rotation  he  understood  a  rotation  relative 
to  the  fixed  stars,  and  here  centrifugal  forces  can 
always  be  found.  "But  how  we  are  to  collect," 
says  Newton  in  the  Scholium  at  the  end  of  the 
Definitions,  ''the  true  motions  from  their  causes, 
effects,  and  apparent  differences,  and  vice  versa  ; 
how  from  the  motions,  either  true  or  apparent,  we 
may  come  to  the  knowledge  of  their  causes  and 
effects,  shall  be  explained  more  at  large  in  the 
following  Tract."  The  resting  sphere  of  fixed  stars 
seems  to  have  made  a  certain  impression  on  Newton 
as  well.  The  natural  system  of  reference  is  for 
him  that  which  has  any  uniform  motion  or  trans- 
lation without  rotation  (relatively  to  the  sphere  of 
fixed  stars).  ^  But  do  not  the  words  quoted  in 
inverted  commas  give  the  impression  that  Newton 
was  glad  to  be  able  now  to  pass  over  to  less  pre- 
carious questions  that  could  be  tested  by  experience  ? 

XIX 

[Instead  of  line  4  up  of  p.  232  to  line   18  of  p.  233, 
put:] 

When  Newton  examined  the  principles  of 
mechanics  discovered  by  Galileo,  the  great  value  of 
the  simple  and  precise  law  of  inertia  for  deductive 

^  Principia,  p.  19,  Coroll.  V  :  "The  motions  of  bodies  included  in 
a  given  space  are  the  same  among  themselves,  whether  that  space  is  at 
rest  or  moves  uniformly  forwards  in  a  right  line  without  any  circular 
motion." 


ADDITIONS  AND  ALTERATIONS      35 

derivations  could  not  possibly  escape  him.  He 
could  not  think  of  renouncing  its  help.  But  the 
law  of  inertia,  referred  in  such  a  naive  way  to  the 
earth  supposed  to  be  at  rest,  could  not  be  accepted 
by  him.  For,  in  Newton's  case,  the  rotation  of  the 
earth  was  not  a  debatable  point  ;  it  rotated  without 
the  least  doubt.  Galileo's  happy  discovery  could 
only  hold  approximately  for  small  times  and  spaces, 
during  which  the  rotation  did  not  come  into  question. 
Instead  of  that,  Newton's  conclusions  about  planetary 
motion,  referred  as  they  were  to  the  fixed  stars, 
appeared  to  conform  to  the  law  of  inertia.  Now, 
in  order  to  have  a  generally  valid  system  of  re- 
ference, Newton  ventured  the  fifth  corollary  of  the 
Principia  (p.  19  of  the  first  edition).  He  imagined 
a  momentary  terrestri^.1  system  of  co-ordinates,  for 
which  the  law  of  inertia  is  valid,  held  fast  in  space 
without  any  rotation  relatively  to  the  fixed  stars. 
Indeed  he  could,  without  interfering  with  its  use- 
abihty,  impart  to  this  system  any  initial  position 
and  any  uniform  translation  relatively  to  the  above 
momentary  terrestrial  system.  The  Newtonian  laws 
of  force  are  not  altered  thereby  ;  only  the  initial 
positions  and  initial  velocities — the  constants  of  in- 
tegration— may  alter.  By  this  view  Newton  gave 
the  exact  meaning  of  his  hypothetical  extension  of 
Galileo's  law  of  inertia.  We  see  that  the  reduction 
to  absolute  space  was  by  no  means  necessary,  for  the 
system  of  reference  is  just  as  relatively  determined 


36        THE  SCIENCE  OF  MECHANICS 

as  in  every  other  case.  In  spite  of  his  metaphysical 
liking  for  the  absolute,  Newton  was  correctly  led  by 
the  tact  of  the  natural  investigator.  This  is  particu- 
larly to  be  noticed,  since,  in  former  editions  of  this 
book,  it  was  not  sufficiently  emphasised.  How  far 
and  how  accurately  the  conjecture  will  hold  good  in 
future  is  of  course  undecided. 


XX 

[To  p.  238,  line  3,  add  :] 

I  do  not  believe  that  the  writings  of  the  advocates 
of  absolute  space  which  have  appeared  during  the 
last  ten  years  can  assert  anything  else  than  the 
italicised  passage,  which  stood  in  the  first  German 
edition  of  1883  (pp.  221,  222), 

XXI 

[Appendix  XX,  on  pp.  542-547,  is,  in  the  seventh 
German  edition,  partly  omitted,  and  the 
following  inserted  :] 

The  law  of  inertia  has  often  been  discussed  in 
ancient  and  modern  times,  and  almost  always  the 
empty  conception  of  absolute  space,  which  is  open 
to  such  grave  objections  in  point  of  principle,  has 
mixed  itself  up  with  it  in  a  disturbing  manner. 
Here  we  will  limit  ourselves  to  the  mention  of  the 
more  modern  discussions  of  this  subject. 


ADDITIONS  AND  ALTERATIONS      37 

In  the  first  place  we  must  mention  the  writings 
of  C.  Neumann  :  Ueber  die  Principien  der  Galilei- 
Newton' scJien  Theorie,  of  1870,  and  "IJber  den 
Korper  Alpha "  {Be7'.  der  konigl,  sacks.  Ges.  der 
Wiss.y  1 9 10,  iii).  The  author  denotes,  on  p.  22 
of  the  former  treatise,  the  relation  to  the  body 
Alpha  as  a  relation  to  a  system  of  axes  which 
proceeds  uniformly  in  a  straight  line  without  rota- 
tion, and  thus  his  statement  coincides  with  the  fifth 
corollary  of  Newton  which  we  have  already  men- 
tioned. However,  I  do  not  believe  that  the  fiction 
of  the  'body  Alpha  and  the  preservation  of  the  dis- 
tinction between  absolute  and  relative  motion  and 
the  paradoxes  (pp.  27,  28)  connected  with  this  dis- 
tinction have  particularly  contributed  to  the  clari- 
fication of  the  matter.  In  the  publication  of  1910 
(p.  70,  note  i)  Neumann  calls  what  he  has  brought 
forward  purely  hypothetical,  and  in  this  lies  an 
essential  progress  in  the  knowledge  of  Newton's 
fifth  corollary.  In  the  same  publication,  Lange's 
standpoint  is  exposed  as  in  essentials  coinciding 
with  his  own. 

H.  Streintz  (Die  physikalischen  Grundlagen  der 
Mechanik,  1883)  accepts  the  Newtonian  distinction 
between  absolute  and  relative  motion,  but  also 
comes  to  the  view  expressed  in  Newton's  fifth 
corollary.  What  I  had  to  say  against  Streintz's 
criticism  of  my  views  was  contained  in  the  former 
editions  of  this  work  and  shall  not  be  repeated  here. 


38         THE  SCIENCE  OF  MECHANICS 

We  will  now  consider  L.  Lange  :  '*i)ber  die 
wissenschaftliche  Fassung  der  Galilei'schen  Behar- 
rungsgesetzes,"  Wundt's  P kilos.  Studien,  vol.  ii, 
1885,  pp.  266-297,  539-545  ;  Ber.  d.  konigl.  sacks. 
Ges.  der  Wz'ss.,  7natk.-pkysik.  Klasse,  1885,  pp.  333- 
351  ;  Die geschicktlicke  Entwicklung des  Bewegungs- 
begriffs^  Leipsic,  1886;  Das  Inertialsystem  vor  dem 
Forum  der  Naturforsckung^  Leipsic,  1902. 

L.  Lange  sets  out  from  the  supposition  that  the 
general  Newtonian  law  of  inertia  subsists  and  seeks 
the  system  of  co-ordinates  to  which  it  is  to  be  referred 
(1885).  With  respect  to  any  moving  point  1\ — 
which  can  even  move  in  a  curve, — we  can  so  move 
a  system  of  co-ordinates  that  the  point  \\  describes 
a  straight  line  G^  in  this  system.  If  we  have  also 
a  second  moving  point  \\,  the  system  can  still  be 
moved  so  that  a  second  straight  line  G2,  in  general 
warped  with  respect  to  G^,  is  described  by  Pg,  if 
only  the  shortest  distance  G^  G^  does  not  surpass  the 
shortest  distance  which  \\  P.^  can  ever  have.  Still  the 
system  can  rotate  about  P^  Pg.  If  we  choose  a  third 
straight  line  G3,  such  that  all  the  triangles  \\,V2,Y^ 
which  can  arise  by  means  of  any  third  moving  point 
P3  are  representable  by  points  on  G^.G^,^^,  then 
P3  can  also  advance  on  G3.  Thus,  for  at  most  three 
points,  a  system  of  co-ordinates  in  which  these  points 
proceed  in  a  straight  line  is  a  mere  convention. 
Now,  Lange  sees  the  essential  contents  of  the  law 
of   inertia    in    that,   by  the   help  of   three  material 


ADDITIONS  AND  ALTERATIONS      39 

points  which  are  left  to  themselves,  a  system  of 
co-ordinates  can  be  found  with  respect  to  which  four 
or  arbitrarily  many  material  points  which  are  left 
to  themselves  move  in  a  straight  line  and  describe 
paths  which  are  proportional  to  one  another.  The 
process  in  nature  is  thus  a  simplification  and  limi- 
tation of  the  kinematically  possible  variety  of 
cases. 

This  promising  thought  and  its  consequences  found 
much  recognition  with  mathematicians,  physicists, 
and  astronomers.  {Cf.  H.  Seeliger's  account  of 
Lange's  works  in  the  Vierteljahrsschrift  der  astro- 
no77i.  Ges.,  vol.  xxii,  p.  252;  H.  Seeliger,  ''Uber 
die  sogenannte  absolute  Bewegung,"  Sitzungsber. 
der  Miinchener  Akad.  der  Wzss.,  1906,  p.  85.) 
Now,  J.  Petzoldt  (''Die  Gebiete  der  absoluten  und 
der  relativen  Bewegung,"  Ostwald's  Annalen  der 
Naturphilosophie,  vol.  vii,  1908,  pp.  29-62)  has 
found  certain  difficulties  in  Lange's  thoughts,  and 
these  difficulties  have  also  disturbed  others  and  are  not 
quickly  to  be  put  on  one  side.  On  this  account  we 
will  here  break  off  our  remarks  on  Lange's  system 
of  co-ordinates  or  inertial  systems  till  the  clouds  pass 
away.  Seeliger  has  attempted  to  determine  the 
relation  of  the  inertial  system  to  the  empirical 
astronomical  system  of  co-ordinates  which  is  in  use, 
and  believes  that  he  can  say  that  the  empirical 
system  cannot  rotate  about  the  inertial  system  by 
more  than  some  seconds  of  arc  in  a  century.      Cf. 


40         THE  SCIENCE  OF  MECHANICS 

also  A.  Anding,  "  Cber  Koordinaten  und  Zeit," 
in  vol.  vi  of  the  Encyklopddie  der  mathematischen 
Wissefischaften. 

The  view  that  "  absolute  motion"  is  a  conception 
which  is  devoid  of  content  and  cannot  be  used  in 
science  struck  almost  everybody  as  strange  thirty 
years  ago,  but  at  the  present  time  it  is  supported 
by  many  and  worthy  investigators.  Some  "re- 
lativists" are:  Stallo,  J.  Thomson,  Ludwig  Lange, 
Love,  Kleinpeter,  J.  G.  MacGregor,  Mansion, 
Petzoldt,  Pearson.  The  number  of  relativists  has 
very  quickly  grown,  and  the  above  list  is  certainly  in- 
complete. Probably  there  will  soon  be  no  important 
supporter  of  the  opposite  view.  But,  if  the  incon- 
ceivable hypotheses  of  absolute  space  and  absolute 
time  cannot  be  accepted,  the  question  arises  :  In 
what  way  can  we  give  a  comprehensible  meaning 
to  the  law  of  inertia  ?  MacGregor  shows  in  an 
excellent  paper  {Phil.  Mag.,  vol.  xxxvi,  1893, 
pp.  233-264),^  which  is  very  clearly  written  and 
shows  great  recognition  of  Lange's  work,  that  there 
are  two  ways  that  we  can  take  :  (i)  the  historical 
and  critical  way,  which  considers  anew  the  facts  on 
which  the  law  of  inertia  rests  and  which  draws  its 
limits  of  validity  and  finally  considers  a  new  formu- 
lation ;  (2)  the  supposition  that  the  law  of  inertia 
in   its  old  form  teaches   us  the  motions  sufficiently, 

^  [This  paper,  "  On  the  Hypotheses  of  Dynamics,"  was  occasioned 
by  some  remarks  of  O.  Lodge  on  a  former  paper  of  MacGregor's.] 


ADDITIONS  AND  ALTERATIONS      41 

and    the    derivation    of  the    correct    system   of   co- 
ordinates from  these  motions. 

P'or  the  first  method  it  seems  to  me  that  Newton 
himself  gave  the  first  example  with  his  system  of 
reference  indicated  in  the  fifth  corollary,  which  has 
been  often  mentioned  above.  It  is  obvious  that  we 
must  take  account  of  modifications  of  expression 
which  have  become  necessary  by  extension  of  our 
experience.  The  second  way  is  very  closely  con- 
nected psychologically  with  the  great  trust  which 
mechanics,  as  the  most  exact  natural  science,  enjoys. 
Indeed,  this  way  has  often  been  followed  with  more 
or  less  success.  W.  Thomson  and  P.  G.  Tait 
{Treatise  on  Natural  P  kilo  sop  J ly,  vol.  i,  part  i,  1879, 
§  249)^  remark  that  two  material  points  which  are 
simultaneously  projected  from  the  same  place  and 
then  left  to  themselves  move  in  such  a  way  that 
the  line  joining  them  remains  parallel  to  itself. 
Thus,  if  four  points  O,  P,  Q,  and  R  are  projected 
simultaneously  from  the  same  place  and  then  subject 
to  no  further  force,  the  lines  OP,  00,  and  OR 
always  give  fixed  directions.  J.  Thomson  attempts, 
in  two  articles  {Proc.  Roy.  Soc.  Edinb.^  1884, 
pp.  568,  730),  to  construct  the  system  of  reference 
corresponding  to  the  law  of  inertia,  and  in  this 
recognises  that  the  suppositions  about  uniformity 
and  rectilinearity  3.yq  partly  conventional.  Tait  {loc. 
cit.^  p.   743),  stimulated  by  J.   Thomson,  takes  part 

'  [C/.  §§267,  245.] 


42        THE  SCIENCE  OF  MECHANICS 

in  the  solution  of  the  same  problem  by  quaternions. 
We  find  also  MacGregor  in  the  same  path  ("The 
Fundamental  Hypotheses  of  Abstract  Dynamics," 
Trans.  Roy.  Soc.  of  Canada,  vol.  x,  1892,  §  iii, 
especially  pp.  5  and  6). 

The  same  psychological  motives  were  certainly 
active  in  the  case  of  Ludwig  Lange,  who  has  been 
most  fortunate  in  his  efforts  correctly  to  interpret 
the  Newtonian  law  of  inertia.  This  he  did  in  two 
articles  in  VVundt's  PJiilos.  Studien  of  1885. 

More  recently  Lange  {PJiilos.  Studien,  vol.  xx, 
1902)  published  a  critical  paper  in  which  he  also 
worked  out  the  method  of  obtaining  a  7iew  system 
of  co-ordinates  according  to  his  principles,  when  the 
usual  rough  reference  to  the  fixed  stars  shall  be,  in 
consequence  of  more  accurate  astronomical  observa- 
tions, no  longer  sufficient.  There  is,  I  think,  no 
difference  of  meaning  between  Lange  and  myself 
about  the  theoretical  and  formal  value  of  Lange's 
expressions,  and  about  the  fact  that,  at  the  present 
time,  the  heaven  of  fixed  stars  is  the  only  practically 
usable  system  of  reference,  and  about  the  method 
of  obtaining  a  new  system  of  reference  by  gradual 
corrections.  The  difference  which  still  subsists, 
and  perhaps  will  always  do  so,  lies  in  the  fact  that 
Lange  approaches  the  question  as  a  mathematician, 
while  I  was  concerned  with  the  physical  side  of  the 
subject. 

Lange    supposes    with   some  confidence  that   his 


ADDITIONS  AND  ALTERATIONS      43 

expression  would  remain  valid  for  celestial  motions 
on  a  large  scale.  I  cannot  share  this  confidence. 
The  surroundings  in  which  we  live,  with  their 
almost  constant  angles  of  direction  to  the  fixed 
stars,  appear  to  me  to  be  an  extremely  special  case, 
and  I  would  not  dare  to  conclude  from  this  case  to 
a  very  different  one.  Although  I  expect  that  astro- 
nomical observation  will  only  as  yet  necessitate 
very  small  corrections,  I  consider  it  possible  that 
the  law  of  inertia  in  its  simple  Newtonian  form  has 
only,  for  us  human  beings,  a  meaning  which  depends 
on  space  and  time.  Allow  me  to  make  a  more 
general  remark.  We  measure  time  by  the  angle 
of  rotation  of  the  earth,  but  could  measure  it  just 
as  well  by  the  angle  of  rotation  of  any  other  planet. 
But,  on  that  account,  we  would  not  believe  that  the 
temporal  course  of  all  physical  phenomena  would 
have  to  be  disturbed  if  the  earth  or  the  distant 
planet  referred  to  should  suddenly  experience  an 
abrupt  variation  of  angular  velocity.  We  consider 
the  dependence  as  not  immediate,  and  consequently 
the  temporal  orientation  as  external.  Nobody  would 
believe  that  the  chance  disturbance — say  by  an 
impact — of  one  body  in  a  system  of  uninfluenced 
bodies  which  are  left  to  themselves  and  move  uni- 
formly in  a  straight  line,  supposing  that  all  the 
bodies  combine  to  fix  the  system  of  co-ordinates, 
will  immediately  have  a  disturbance  of  the  others 
as_^consequence.       The  orientation  is  external  here 


/|4         THE  SCIENCE  OF  MECHANICS 

also.  Although  we  must  be  very  thankful  for  this, 
especially  when  it  is  purified  from  meaninglessness, 
still  the  natural  investigator  must  feel  the  need  of 
further  insight — of  knowledge  of  the  immediate  con- 
nections, say,  of  the  masses  of  the  universe.  There 
will  hover  before  him  as  an  ideal  an  insight  into 
the  principles  of  the  whole  matter,  from  which  ac- 
celerated and  inertial  motions  result  in  the  same  way. 
The  progress  from  Kepler's  discovery  to  Newton's 
law  of  gravitation,  and  the  impetus  given  by  this 
to  the  finding  of  a  physical  understanding  of  the 
attraction  in  the  manner  in  which  electrical  actions 
at  a  distance  have  been  treated,  may  here  serve  as 
a  model.  We  must  even  give  rein  to  the  thought 
that  the  masses  which  we  see,  and  by  which  we  by 
chance  orientate  ourselves,  are  perhaps  not  those 
which  are  really  decisive.  On  this  account  we  must 
not  underestimate  even  experimental  ideas  like  those 
of  Friedlander^  and  Foppl,^  even  if  we  do  not  yet 
see  any  immediate  result  from  them.  Although  the 
investigator  gropes  with  joy  after  what  he  can 
immediately  reach,  a  glance  from  time  to  time  into 
the  depths  of  what  is  uninvestigated  cannot  hurt 
him. 

A    small    elementary    paper    of    J.    R.     Schiitz 
C'Prinzip  der    absoluten   Erhaltung  der   Energie," 

^  B.  and  J.  Friedlander,  Absolute  und  relative  Bewegung,  Berlin, 
1896.    . 

-  "  Uber  einen  Kreiselversuch  zur  Messung  der  Umdrehungsgesch- 
windigkeit  der  Erde,"  Sitzungsber.  der  Miiitchener  Akad.,  1904,  p.  5; 
"  ijber  absolute  und  relative  Bewegung,"  ibid.,  1904,  p.  383. 


ADDITIONS  AND  ALTERATIONS      45 

Gottinger  Nachnchten,  matJi.-physik.  Klasse^  1897) 
shows,  on  simple  examples,  that  Newton's  laws  can 
be  obtained  from  the  principle  spoken  of.  The  term 
"absolute"  is  only  meant  to  express  that  the 
principle  is  to  be  freed  from  an  indeterminateness 
and  arbitrariness.  If  we  imagine  the  principle 
applied  to  the  central  impact  of  elastic  masses 
Wj  and  Wj  ^"  th^  form  of  points,  of  initial  velocities 
u^  and  U2  and  final  velocities  v^  and  z^g?  we  have 

We  c^n  calculate  v^  and  v^  from  u^  and  u^^  if  we 
suppose  that  the  principle  of  energy  holds  for  any 
velocity  of  translation  c  directed  in  the  same  sense 
as  u  and  v.      We  then  have 

If  we  subtract  the  first  equation  from  the  second,  we 
get  the  equation  of  the  principle  of  reaction  : 

in  which  c  has  dropped  out.  From  the  first  and 
third  equation  we  can  calculate  v^  and  v^.  By  an 
analogous  treatment  of  the  '' absolute  "  principle  of 
energy,  we  get  Newton's  equation  of  force  for  a 
mass-point,  and  finally  the  law  of  reaction,  with  its 
corollaries  of  the  conservation  of  the  quantity  of 
motion  and  the  conservation  of  the  centre  of  gravity. 
The  study  of  this  paper  is  very  much  to  be  recom- 
mended,  since  even  the  conception  of  mass  can  be 


46        THE  SCIENCE  OF  MECHANICS 

derived  by  the  help  of  the  principle  of  energy.  Cf, 
the  section  on  "  Retrospect  of  the  Development  of 
Dynamics  "  in  my  Mechanics, 


XXII 

[To  p.  242,  line  6  up,  add:] 

What  is  pleonastic  and  tautological  in  Newton's 
propositions  is  psychologically  comprehensible  if  we 
imagine  an  investigator  who,  setting  out  from  his 
familiar  ideas  of  statics,  is  in  the  act  of  establishing 
the  fundamental  propositions  of  dynamics.  At  one 
time  force  is  in  the  focus  of  consideration  as  a  pull 
or  a  pressure,  and  at  another  time  as  determinative 
of  accelerations.  When,  on  the  one  hand,  he  recog- 
nises, by  the  idea  of  a  pressure  which  is  common  to 
all  forces,  that  all  forces  also  determine  accelera- 
tions, then  this  twofold  notion  leads  him,  on  the 
other  hand,  to  a  divided  and  far  from  unitary  repre- 
sentation of  the  new  fundamental  propositions.  Cf. 
Erkenntnis  ujid  Irrtuvi^  2nd  ed. ,  pp.   140,  315. 

XXIII 
[To  p.  243,  last  line,  add  :] 

The  theorems  a  to  e  were  given  in  my  note 
"Uber  die  Definition  der  Masse"  in  Carl's  Reper- 
toriuni  der  Experimentalphy sik^  vol.  iv,  1868  ;  re- 
printed  in    Erhaltung  der  Arbeit^    1872,    2nd   ed. , 


ADDITIONS  AND  ALTERATIONS      47 

Leipsic,    1909.^     Cf.    also   Poincare,   La  Science   et 
thypothese^  Paris,  pp.   wo  et  seqq. 

XXIV 

[To  p.  245,  ''Tait."] 

On  p.  243  of  the  seventh  German  edition  there  is 
only  mention  of  ''  W.  Thomson  (Lord  Kelvin)." 

XXV 

[To  p.  245,  beginning  of  VIII,  add  :] 

Dynamics  has  developed  in  an  analogous  way  to 
statics.  Different  special  cases  of  motions  of  bodies 
were  observed,  and  people  tried  to  put  these  observa- 
tions in  the  form  of  rules.  But  just  as  little  as, 
from  the  observation  of  a  case  of  equilibrium  of 
the  inclined  plane  or  the  lever,  can  be  derived  a 
mathematically  exact  and  generally  valid  rule  for 
equilibrium  —  on  account  of  the  inaccuracy  of 
measurement, — so  little  can  the  corresponding  thing 
be  done  for  cases  of  motion.  Observation  only 
leads,  in  the  first  place,  to  the  conjecturing  of 
laws  of  motion,  which,  in  their  special  simplicity  and 
accuracy,  are  presupposed  as  hypotheses  in  order  to 
try  whether  the  behaviour  of  bodies  can  be  logically 
derived  from  these  hypotheses.      Only  if  these  hypo- 

^  English  translation  by  Philip  E.  B.  Jouidain  under  the  title  History 
and  Root  of  the  Principle  of  the  Conserz>atiofi  of  Energy,  Chicago  and 
London,  The  Open  Court  Publishing  Company,  191 1.  The  reprint 
referred  to  is  given  on  pp.  80-85  o^  ^^'^  translation. 


48         THE  SCIENCE  OF  MECHANICS 

theses  have  shown  themselves  to  hold  good  in  many 
simple  and  complicated  cases,  do  we  agree  to  keep 
them.  Foincare,  in  his  La  science  et  Vhypothcse,  is, 
then,  right  in  calling  the  fundamental  propositions 
of  mechanics  conventions  which  might  very  well 
have  fallen  out  otherwise. 

XXVI 

[On  p.  247,  line  17,  insert  :] 

He  believed  that  he  could  conclude  from  this  the 
proportionality  of  the  spaces  fallen  through  with  the 
squares  of  the  times  of  falling  {Ediz.  Nazionale, 
vol.  viii,  pp.  ni,  374). 

XXVII 
[To  p.  248,  line  10  up,  add  :] 

In  the  second  infinitesimal  supposition  of  Galileo 
— of  proportionality  of  the  velocity  to  the  time  of 
falling — the  triangular  surfaces  of  Galileo's  con- 
struction (fig.  87  of  my  Mechanics)  represent,  in  a 
beautiful  and  intuitive  way,  the  paths  that  are 
described.  With  the  first  supposition,  on  the  other 
hand,  the  analogous  triangles  have  no  phoronomical 
signification,  and  on  this  account  the  integration  was 

not  successful. 

XXVIII 

[On  p.  255,  line  12,  add  :] 

It  has  been  shown  that  the  present  form  of  our 

science  of  mechanics  rests  on  a  historical   accident. 


ADDITIONS  AND  ALTERATIONS      49 

This  is  also  shown  in  a  very  instructive  way  by  the 
remarks  of  Lieut. -Col.  Hartmann  in  his  paper  on 
the  *' Definition  physique  de  la  force"  (Congres 
international  de  philosophie^  Geneva,  1905,  p.  728), 
and  in  L'enseignement  mat/ientatique,  Paris  and 
Geneva,  1904,  p.  425.  The  author  shows  the  use 
of  the  usual  conceptions  of  different  ideas. 

XXIX 

To  note  on  p.  555  :  "It  should  be  added  that  a 
second  edition  of  Die  Geschichte  und  die  Wurzel  des 
^atzes  von  der  Erhaltung  der  Arbeit  appeared  at 
Leipsic  in  1909,  and,  as  already  mentioned,  an 
English  translation,  under  the  title  History  and  Root 
of  the  Principle  of  the  Conservation  of  Energy ^  was 
published  at  Chicago  and  London  in  191 1." 

The  same  remark  applies  to  the  notes  on  pp.  494, 
496,  and  567,  and  the  text  on  pp.  580  and  585. 

XXX 

The  passage  on  p.  571,  line  10  up  of  text,  to 
p.  572,  line  2,  is  omitted  in  the  seventh  German 
edition,  and  the  following  added:  ''It  must  here 
again  be  emphasised  that  Newton,  in  his  fifth 
corollary,  often  quoted  above,  and  which  alone  has 
scientific  value,  does  not  make  absolute  space  his 
system  of  reference." 

On  p.  572,  the  last  sentence  in  the  text  is 
omitted,   and  the  sentence  added:   *'But   that   the 

4 


50        THE  SCIENCE  OF  MECHANICS 

world  is  without  influence  must  not  be  supposed  in 
advance.  In  fact,  Neumann's  paradoxes  only  vanish 
when  we  give  up  absolute  space  and  do  not  go 
beyond  the  fifth  corollary." 


XXXI 

[To  p.  302,  after  line  19,  add  :] 

A.  Schuster  of  Manchester  has  proved  in  a  very 
beautiful  way,  in  the  London  Philosophical  Trans- 
actions for  1876  (vol.  clxvi,  p.  715),  that  the  forces 
which  set  the  radiometer  of  Crookes  and  Geissler  in 
motion  are  inner  forces.  If  we  put  the  vanes  of  the 
radiometer  into  rotation  by  means  of  light,  after  we 
have  suspended  the  glass  cover  bifilarly,  this  cover 
immediately  shows  a  tendency  to  rotate  in  a  sense 
contrary  to  the  vanes.  Schuster  was  able  to 
measure  the  magnitude  of  the  forces  which  here 
came  into  action. 

V.  Dvorak  of  Agram,  the  discoverer  of  the 
acoustic  reaction-wheel,  has,  at  my  request,  carried 
out  analogous  experiments  with  his  reaction-wheel. 
If  we  put  the  resonator-wheel  into  acoustical  rota- 
tion, its  light  cylindrical  glass  cover,  which  floated 
on  water,  fell  at  once  into  rotation  in  the  opposite 
sense,  and  this  latter  rotation,  when  the  wheel  only 
goes  on  rotating  by  inertia,  also  immediately  re- 
verses its  sense  of  rotation.  My  son,  Ludwig  Mach, 
has,   at  my  wish,   improvised  upon  the  experiment 


ADDITIONS  AND  ALTERATIONS      51 

with  Dvorak's  wheel  by  replacing  the  glass  cover  by 
a  light  paraffined  paper  cover  which  floated  on  water. 
When  such  a  paper  cover  was  suspended  bifilarly, 
every  acceleration  of  the  wheel  showed  an  increased 
tendency  to  rotation  in  the  opposite  sense,  and  every 
retardation  a  diminished  tendency  of  this  kind  ;  and 
this  was  shown  in  a  very  striking  manner.  Dvorak's 
experiments  are  explained  by  those  with  the  motor 
represented  in  fig.  152  of  my  Mechanics^  and,  in 
especial,  by  the  experiment  of  fig.  153^.  Cf. 
A.  Haberditzl,  "  Uber  kontinuierliche  akustische 
Rotation  und  deren  Beziehung  zum  Flachenprinzip," 
Sitzungsber.der  Wiener  AkademiCy  math.-naturwiss. 
Klasse,  May  9th,  1878. 

XXXII 

The  passage  *'  In  spite  .  .  .  reasonable  con- 
jecture "  on  pp.  305-308  and  note  on  p.  308  of 
the  Mechanics  is  omitted  in  the  seventh  German 
edition,  and  the  passage  added:  ''According  to 
Wohlwill's  researches  {Zeitschrifi  fur  Volkerpsyc/to- 
logie,  vol.  XV,  1884,  p.  387),  Marci  emphatically 
cannot  be  regarded  as  having  advanced  dynamics 
in  the  direction  taken  by  Galileo." 

XXXIII 

[To  p.  364,  line  21,  add  :] 

The  paper  of  Lipschitz  ('*  Bemerkungen  zu  dem 
Prinzip  des  kleinsten  Zwanges,"/c«r«^////>  il/«///. , 


52        THE  SCIENCE  OF  MECHANICS 

vol.    Ixxxii,    1877,   PP-    316  et  seqq.)  contains   pro- 
found   investigations    on    the    principle    of    Gauss. 
Many    elementary    examples,    on    the    other    hand, 
are  to  be  found  in   K.  HoUefreund's  Anwendungen 
des     Gauss'scJien    Prinzips    vom    kleinsten    Zwange 
(Berlin,    1897).      On    the    principle    here  spoken  of 
and    allied    principles,     see     Ostwald's     Klassiker^ 
No.    167  :    AbJiandlu7ige7i    uber   die    Prinzipien    der 
Mechanik    vofi    Lagrange^    Rodrigues,    Jacobi    und 
Gauss^   edited    by   Philip   E.    B.   Jourdain  (Leipsic, 
1908).      The    notes   of  Jourdain    on    pp.    31-68   go 
beyond  the  needs  of  a  first  orientation,  and  this  orien- 
tation is  the  object  of  the  present  elementary  book. 
What  is  said  on  pp.    363-364  of  my  Mechanics 
stands  in  need  of  completion.      If  the  masses  of  the 
system    have   no  velocity,   the  actual  motions  only 
enter  in  the  sense  of  possible  work,  which  is  con- 
sistent with  the  conditions  of  the  system  (C.  Neu- 
mann, Ber.  der  kgl.  sacks.  Ges.  der  Wiss.,  vol.  xliv, 
1892,   p.    184).      But  if  the  masses  have  velocities, 
which  can  even  be  directed  against  the  impressed 
forces,  then  the  motions  which  are  determined  by 
the  velocities  and  forces  are  superposed  (Boltzmann, 
Ann.  der  Phys.   und  Chem,^  vol.  Ivii,  1896,  p.  45), 
and    Ostwald's    maximum-principle    {Lehrbuch   der 
allgem.     Chemie^    vol.    ii,    part   i,    1892,    p.    37)   is, 
according    to    Zemplen's    excellent   and   universally 
comprehensible  remark  (Ann.  der  Phys.  und  Cheni.^ 
vol.  X,  1903,  p.  428),  unsuitable  for  the  description  of 


ADDITIONS  AND  ALTERATIONS      53 

mechanical  events,  because  it  does  not  take  account 
of  the  inertia  of  the  masses.  However,  it  remains 
correct  that  the  (virtual)  works  which  are  consistent 
with  the  conditions  become  actual.  My  text,  which 
was  drawn  up  before  1882,  could  not,  of  course, 
take  account  of  the  attempts  to  found  an  energetical 
mechanics  of  two  years  later.  For  the  rest,  I  cannot 
value  these  attempts  so  little  as  some  do.  Even 
the  old  "classical"  mechanics  has  not  arrived  at 
its  present  form  without  passing  through  analogous 
stages  of  error.  In  particular.  Helm's  view  {Die 
Energeiik  nach  Hirer  geschichtlichen  Entwickelung^ 
Leipsic,  1898,  pp.  205-252)  can  hardly  be  objected 
to.  Cf.  my  exposition  of  the  equal  justification  of 
the  conceptions  of  work  and  force  {Ber,  der  Wiener 
Akad.^  December  1873),  and  also  many  passages  of 
my  Mechanics,  particularly  pp.  248  et  seqq. 

XXXIV 

[On  p.  575,  line  6  up,  insert :] 

So  much  was  already  laid  down  in  the  first 
German  edition  of  1883.  The  objection  of  Helm 
{Energetik,  p.  247),  in  so  far  as  it  concerns  my  own 
work,  is  hardly  just. 

XXXV 

[To  p.  481,  line  i,  add  :] 

Cf.  my  paper,  '*Die  Leitgedanken  meiner  natur- 
wissenschaftlichen    Erkenntnislehre    und    ihre    Auf- 


54         THE  SCIENCE  OF  MECHANICS 

nahme  durch  die  Zeitgenossen "  {Scientia :  Rivista 
di  Scienza,  vol.  vii,  19 lO,  No.  14,  2  ;  or  Physik- 
alische  Zeitschrift,  19 10,  pp.  599-606). 

XXXVI 

[To  p.  504,  end  of  §  i,  add  from  Mechanik,  p.  480  :] 

With  these  lines,  which  were  written  in  1883, 
compare  Petzoldt's  remarks  on  the  striving  after 
stability  in  intellectual  life  ("  Maxima,  Minima 
und  Okonomie,"  Viei'teljahrsschr.  fiir  wiss.  Phil- 
osophies   1 89 1 ). 

XXXVII 

[To  p.   507,  last  line,  add  :] 

CONCLUSION 

At  the  beginning  of  this  book,  the  view  was 
expressed  that  the  doctrines  of  mechanics  have 
developed  out  of  the  collected  experiences  of  handi- 
craft by  an  intellectual  process  of  refinement.  In 
fact,  if  we  consider  the  matter  without  prejudice,  we 
see  that  the  savage  discoverers  of  bow  and  arrows, 
of  the  sling,  and  of  the  javelin,  set  up  the  most 
important  law  of  modern  dynamics — the  law  of 
inertia — long  before  it  was  misunderstood  with 
thorough-going  perversity  by  Aristotle  and  his 
learned  commentators.  And  although  first  ancient 
machines  for  throwing  projectiles  and  catapults  and 
then  modern  firearms  brought  this  law  daily  before 


ADDITIONS  AND  ALTERATIONS      55 

our    eyes,   many  centuries   were  needed  before  the 
correct  theoretical  idealisation  was  discovered  by  the 
genius  of   Galileo  and  Newton.     It  lay  in  exactly 
the    opposite    direction  to  that  in  which  the  great 
majority  of  human  beings  expected  it  to  lie.      Not  the 
conservation,  but  the  decrease  of  the  velocity  of  pro- 
jection was  to  be  theoretically  explained  and  justified. 
The  simple  machines, — the  five  mechanical  powers 
— as  they  are  described  by  Hero  of  Alexandria  in 
the   work    of   which    an    Arabian    translation    came 
down  to  the  Middle  Ages,   are  without  question  a 
product  of  handicraft.      If,  now,  a  child  busies  him- 
self  with  mechanical    work  with    quite  simple  and 
primitive    means, — as  was  the    case    with    my  son, 
Ludwig   Mach — the  dynamical   sensations  observed 
in  this  connection  and    the    dynamical  experiences 
obtained  when  adaptive  motions  are  made,  make  a 
powerful  and  lasting  impression.     If  we  pay  attention 
to  these  sensations,   we    come  closer,   intellectually 
speaking,  to  the  instinctive  origin  of  the  machines. 
We  understand  why  a  long  lever  which  gives  back 
a  less  pressure  is  preferred,  and  why  a  hammer  which 
is  swung  round  to  the  nail  can  transfer  more  work 
or    vis    viva    to    it.      We    understand    at    once    by 
experiment  the   transport  of   loads   on   rollers,   and 
also  how  the  wheel — the  fixed  roller — arose.      The 
making  of  rollers  must  have  gained  a  great  technical 
importance  and  have    led  to    the    discovery  of   the 
turning-lathe.      In  possession  of  this,  mankind  easily 


56        THE  SCIENCE  OF  MECHANICS 

discovered  the  wheel,  the  wheel  and  axle,  and  the 
pulley.      But  the  primitive  turning-lathe  is  the  very 
ancient  fire-drill  of  savages,  which  had  a  bow  and 
cord,  though  of  course  this  primitive  lathe  is  only 
fitted  for  small  objects.      The  Arabians  still  use  it, 
and,   up  to  quite  recent  times,   it  was  almost    uni- 
versally in  use  with  our  watchmakers.      The  potter's 
wheel  of  the  ancient  Egyptians  was  also  a  kind  of 
turning-lathe.      Perhaps  these  forms  served  as  models 
for  the  larger  turning-lathe,  whose  discovery,  as  well 
as  that  of  the  plumb-line  and  theodolite,  is  ascribed 
to  Theodorus  of  Samos.      On  it  pillars  of  stone  may 
well  have  been  turned  (532  B.C.).      Not  all  pieces  of 
knowledge  find  a  like  use  ;  often  they  lie  fallow  for 
a  long  time.      The  ancient  Egyptians  had  wheels  on 
the  war-chariots  of  the  king.      They  actually  trans- 
ported   their    huge    stone    monuments,   with   brazen 
disregard  of   the  work  of  men,   on  sledges.      What 
did  the  labour  of  slaves  taken  as  prisoners  in  war 
matter  to  them  ?    The  prisoners  ought  to  be  thankful 
that  they  were  not,  in  the  Assyrian  manner,  impaled, 
or  at  least  blinded,  but  only,  quite  kindly,  in  com- 
parison with  that,  used  as  beasts  of  burden.      Even 
our  noble  precursors  in  civilisation — the  Greeks — did 
not  think  very  differently. 

But  if  we  suppose  even  the  best  will  for  progress, 
many  discoveries  remain  hardly  comprehensible. 
The  ancient  Egyptians  were  not  acquainted  with  the 
screw.      In  the  many  plates  of  Rossellini's  work  no 


ADDITIONS  AND  ALTERATIONS       57 

trace  of  it  is  to  be  found.  The  Greeks  ascribed, 
on  doubtful  reports,  its  discovery  to  Archytas  of 
Tarentum  (about  390  B.C.).  But  with  Archimedes 
(250  B.C.)  and  with  Hero  (100  B.C.)  we  find  the 
screw  in  very  many  forms  as  something  well  known. 
Hero  can  easily  say — and  even  in  a  way  that  can 
be  understood  by  modern  schoolmen  :  **the  screw  is 
a  winding  wedge."  But  whoever  has  not  yet  seen  or 
handled  a  screw  will  not  by  this  indication  discover 
one.  By  analogy  with  the  cases  spoken  of  before,  we 
must  suppose  that,  when  an  object  in  the  form  of  a 
screw — such  as  a  twisted  rope  or  a  pair  of  wires  twisted 
together  for  ornamental  purposes  or  the  spindle-ring 
of  an  old  fire-drill  which  had  been  worn  spirally  by 
the  cord — fell,  by  chance,  into  someone's  hand,  the 
thought  of  construction  of  a  screw  lay  near  to  the 
sensation  of  the  twisting  of  this  thing  in  and  out  of 
the  hand.  At  bottom  it  is  chance  observations  in 
which  the  faulty  adaptation  of  human  beings  to  their 
surroundings  expresses  itself,  and  which,  when  they 
are  once  remarked,  gives  rise  to  a  further  adaptation. 
My  son  vividly  describes  how,  in  an  ethnographical 
museum,  the  dynamical  experiences  of  his  youth 
again  vividly  came  to  life  ;  how  they  were  awakened 
again  by  the  perceptible  traces  of  the  work  on  the 
objects  exhibited.  May  these  experiences  be  used 
for  the  finding  of  a  universal  genetic  technology,  and 
perhaps,  by  the  way,  lead  a  little  deeper  into  the 
understanding  of  the  primitive  history  of  mechanics. 


CORRECTIONS  TO  BE  MADE  IN  THE 

MECHANICS 

P.    xi,    lines    13-14;    for    ''metaphysical"    read 
**  speculative." 

P.   14  ;  the  last  paragraph  and  figure  are  omitted 
in  the  seventh  German  edition. 

P.  51,  line  8;  add  to  "  Galileo  "  the  words  ''had       ' 
before  this,  in  1594." 

P.   123,  line  22;  after  "  Guericke  "  add:   "which 
were,  in  part,  demonstrated  as  early  as  1654." 

P.     148,    line    4    up;    for     "  Leibnitzians "    read 
"  Leibnizians. "     Similarly  pp.  250,  270. 

P.     188,    line    3;    instead    of    "mean    distances" 
read  "major  axes." 

P.  206,  line  8  ;  for  "constant"  read  "continual." 

P.  271,  line  19;  for  "  Leibnitz  "  read  "Leibniz." 
Similarly  pp.    272,    274,    275,    276,   425,   426,   449, 

454,  575. 

P.  303,  line  8  ;  after  "is"  add  "as  Prof.  Tumlirz 

did." 

P.  335,  line  16  ;  for  "  17 15  "  is  put  in  the  seventh 
German  edition  "  17 14." 

P.  514,  line  6  up  ;  add  to  ''  giometrie''  the  words 

"of  1900." 

58 


CORRECTIONS   TO  BE  MADE  59 

P.  517,  line  2  up  ;  for  ' '  9  "  read  "  6. " 

P.    523,   end  of  Appendix  VIII  ;   add  reference  : 

''^  Erkeyintnis  U7id  Irrtuni^  2nd  ed. ,  Leipsic,  1906." 
P.  566,  line  8  ;  for  ''  Appelt"  read  "  Apelt." 
P.  568,   line  14;  after  ^'Hofler"  add  '' {loc.  cit., 

pp.   120-164)." 

P.   569,  line  7  up  of  text  ;  for   ''412,  448"  read 

**  412-448." 

P.  576,  line  8  ;  for  "unique"  read  ''specialised." 
P.   580,  line  12  up  ;  for  "  1875  "  read  "  1872." 


NOTES  ON 
MACH'S  MECHANICS 

BY 

PHILIP  E.  B.  JOURDAIN,  M. A. (Cantab.). 


NOTES  ON  MACH'S  MECHANICS 

Mach'S  Mechanics  has  become  the  standard  work 

on  the  history  and    philosophy    of   mechanics,   and 

the  author's  wish  {Mechanics,  p.  xvi)  that  no  changes 

shall  be   made  in  the   original  text  of  his  book  is 

binding     not    only    for    personal    reasons    but    also 

because  the  book  is  now    a    classic.      Still,   careful 

historical  and  critical  research  has  shown  me  that 

there  are  some  errors  in    the   book,    and    that    the 

references  often  needed  to  be  verified,   completed, 

and    supplemented    by    other    references    to    more 

easily  accessible  editions  or  translations.       I    hope 

that    the    result    of   the    rather    laborious    work    of 

annotation    undertaken    in  consequence,   which  has 

been  approved  of  by  Professor  Mach  himself,   will 

make  the  book  even   more  useful  to  teachers  and 

students. 

PHILIP  E.  B.  JOURDAIN. 


P.  XV,  line  13  up:   ^'  Sciences,'^ 

The  part  on  the  laws  of  motion  in  this  book  (see 

Preface,  pp.  viii-ix)  is  7iot  by  Clifford,  but  by  Karl 

Pearson. 

63 


64         THE  SCIENCE  OF  MECHANICS 

P.   II,  line  4  up  : 

On  the  deductions  of  Archimedes  and  GaHleo,  cf. 
Mach,  Conservation  of  Energy^  pp.  65-67.  This 
abbreviation  will  always  be  used  for  the  title  : 
History  and  Root  of  the  Principle  of  the  Conservation 
of  Energy,  translated  from  Mach's  work  :  Die 
GescJdchte  U7id  die  Wurzel  des  Satzes  von  der 
Erhaltung  der  Arbeit  (Prague,  1872;  2nd  ed. , 
Leipsic,  1900),  and  annotated  by  Philip  E.  B. 
Jourdain  (Chicago  and  London,  1911). 

P.  13,  line  9:  ''Lagrange." 

This  is  merely  Lagrange's  account  of  Galileo's 
investigation,  in  the  Mica^iique  analytique  of  1788, 
p.  3  {cf.  CEuvres  de  Lagrange,  vol.  xi,  pp.  2-3). 
Archimedes  had  used  a  similar  consideration  to 
determine  the  centre  of  gravity  of  a  magnitude 
composed  of  two  parabolic  surfaces  {De  planorum 
cequilibriis ,  book  ii,  prop.   i). 

Lagrange  told  Delambre  {cf.  Notice  by  Delambre 
in  CEuvres  de  Lagrange,  vol.  i,  p.  xi)  that  he  wrote 
for  Daviet  de  Foncenex,  among  other  things,  a 
new  theory  of  the  lever  in  the  third  part  of  a 
memoir  by  Foncenex  in  the  Miscellanea  Taurinensia 
for  1759.  In  the  second  volume  (1760-61)  of 
the  Miscellanea  Taurinensia  (pp.  299-322)  is  a 
paper  by  Foncenex  entitled:  "  Sur  les  principes 
fondamentaux  de  la  Mecanique,"  in  the  fourth 
section  (''Du  Levier,"  pp.   319-322)  of  which  is  a 


NOTES  ON  MACH'S  ''MECHANICS''     65 

deduction,  by  a  method  depending  on  the  use  of 
Taylor's  series,  which  certainly  seems  very  like 
Lagrange's  work,  of  the  law  of  the  lever.  As 
regards  physical  principles,  it  starts  from  the  fact 
that  two  forces,  each  equal  to  //2,  have  the  same 
effect  as  one  force  /  applied  halfway  between 
them.  Cf.  my  paper  on  ''The  Ideas  of  the 
Fonctions  Analytiques  in  Lagrange's  Early  Work  " 
in  the  Proceedijigs  of  the  International  Congress  of 
Mathematicians  held  at  Cambridge  in  191 2. 

P.  23,  last  line  : 

Guido   Ubaldi,   in    1577,   applied  the  principle   of 
moments  to  the  theory  of  simple  machines. 

P.  24,  line  3  : 

On  this  work  of  Stevinus's,  cf.  Mach,  Conse7'vation 
of  Energy^  pp.  21-23. 

P.  33,  line  23  : 

Lagrange  {CEuvres^  vol.  xi,  pp.  9-1 1)  refers  also, 
on  the  subject  of  equilibrium  on  the  inclined  plane, 
to  Galileo  {Meca?tique,  first  published  in  French  by 
Father  Mersenne  in  1634)  and  Roberval  {Traite  de 
Mecanique,  printed  in  Mersenne's  Harmonie  uni- 
verselle^  1636). 

P.  39,  Hne  7  :   "first." 

The  composition  of  motions  was  known,  says 
Lagrange,  to  Aristotle  {cf.  some  passages  in  his 
Mechanical  Questions)y  was  used  in  the  description 

5 


66         THE  SCIENCE  OF  MECHANICS 

of  curves  by  Archimedes,  Nicomedes,  and  others 
of  the  ancients,  and  Roberval,  and  was  used  in 
mechanics  by  Galileo  (prop.  2  of  the  fourth  day  of 
his  Dialogues^  and  his  Delle  scienza  meccanica  ;  see 
Mechanics,  pp.  154-155,  526),  Descartes,  Roberval, 
Mersenne,  Wallis,  and  others. 

On  the  objections  to  the  deduction  of  the  parallelo- 
gram of  forces  from  that  of  motions,  see  A.  Voss, 
Encykl.  der  math.    IVtss.,  iv.   i,  pp.  43-44. 

P.  s6,  line  12  :   ''death." 

Pierre  Varignon's  Projet  de  la  nouvelle  mhanique 
appeared  in  1687,  at  Paris,  and  his  Nouvelle 
Micanique  in  1725. 

Newton  ''proved"  (see  Mach's  Mechanics,  p.  242) 
the  theorem  of  the  parallelogram  of  accelerations  as 
a  corollary  of  his  second  law  of  motion. 

P.  36,  line  15  :   "theorem." 

Nouvelle  Mecanique,  Sect,  i,  Lemme  xvi. 

P.  36,  note  : 

On  the  history  of  Varignon's  and  Lami's  dis- 
coveries, see  Lagrange,  CEuvres,  vol.  xi,  pp.  15-17. 

P.  40,  line  13  :   "BernoulH." 

This  paper  by  Daniel  Bernoulli  bears  the  title  : 
"  Examen  Principiorum  Mechanicae,"  and  is  printed 
in  Commeni.  Acad,  Sci.  Imp.  Petrop.,  vol.  i,  1726 
(published  in  1728),  pp.  126-142.  The  proof  was 
simplified  by  d'Alembert  {Opusc.  math.,  vol.  i,  1761  ; 


NOTES  ON  MACEPS  ''MECHANICS''     67 

cf.  vol.  vi,  1773)  and  Aime  {Journ.  de  Math.y 
vol.  i,  1836,  p.  335).  See  Voss,  Encykl.  der  math. 
IViss.,  iv,  I,  p.  44,  note  109  (pp.  44-46  also 
contain  references  to  the  work  of  Poisson  and  others 
in  demonstrating  the  parallelogram  of  forces). 

P.  47,  line  8  up : 

The  English  translation  has  here  misprinted 
' '  Cauchy  "  for  ' '  Varignon. " 

P.  51,  line  7  : 

On  the  principle  of  virtual  velocities  and  its  con- 
nection with  the  principle  of  the  impossibility  of 
a  perpetuuni  mobile^  see  Conservation  of  Energy^ 
pp.  31-32.  Lagrange  said  that  Guido  Ubaldi  is 
the  discoverer  of  the  principle  of  virtual  displace- 
ments. Cf.  also  Voss,  Encykl.  der  math.  Wzss. ,  iv, 
I,  p.  66,  note  180. 

P.  52,  line  13  :  Toricelli." 

jDe  motu  graviumnaturaliter  descendentium^  1664.. 

P.  54,  line  10  :   "  work." 

This  conception  (Galileo's  ''moment")  is  funda- 
mental in  Wallis's  treatment  of  statics  in  his 
MecJianica  of  1670  and  1671.  See  also  the  reference 
to  Descartes  in  Lagrange's  Mecanique. 

P.  56,  line  7  up  :   '*  1717." 

This  letter  was  dated  January  26th,  17 17,  and  was 
printed    at    the   head    of  Section  IX  of  Varignon's 


68         THE  SCIENCE  OF  MECHANICS 

Nouvelle  Mecanique  (vol.  ii,  p.  174).  **  In  every 
equilibrium,"  says  Bernoulli,  "of  any  forces,  in 
whatever  manner  they  are  applied  on  one  another, 
whether  immediately  or  mediately,  the  sum  of  the 
positive  energies  will  be  equal  to  the  sum  of  the 
negative  energies  taken  positively." 

P.  63,  last  line  : 

For  this  example,   cf.    Euler,   Hist  de  I  Acad,   de 
Berlin,   175 1,  p.   193. 

P.  65,  line  10  up  :   ''  zero." 

Fourier   (''Memoire   sur   la  statique,"  Journ.   de 
PEc.  polyt.,  cah.  v,    1798,  pp.    20  et  seqq.\  CEuvres, 
vol.   ii,   pp.   475-521,   in  especial  p.   488)  first  con- 
sidered   the    case    of   the    conditions    in    a    statical 
problem  being  expressed  by  inequalities  instead  of 
equations.      Apparently    independently    of    Fourier 
and   of   one   another,    this   case   was   considered   in 
publications    of    Gauss    (1829)    and     Ostrogradski 
(1838).      Cf.   Voss,  Encykl.  der  math.    Wiss. ,  iv.    i, 
pp.    73-75  ;  and  my  notes  in   Ostw aid's  Klassiker, 
No.  167,  pp.  59-60.     In  connection  with  the  formula- 
tion of  Gauss's  principle  for  inequalities,  see  Voss, 
hoc.  cit.y  pp,  85-87;  Ostwald's  Klassiker,   No.   167, 
pp.  64-65. 

P.  65,  line  9  up  :   "  Mechanics.'' 

In  the  second  and  later  editions  {CEuvres,  vol,  xi, 
pp.  22-26),  not  the  edition  of  1788. 


NOTES  ON  MAC  ITS  ''MECHANICS''     69 

P.  68,  line  19:   "cases." 

On  the  proofs  of  the  principle  of  virtual  displace- 
ments (or  velocities)  of  Fourier  (1798,  and  there- 
fore before  Lagrange's  proof  of  181 1),  Lagrange 
(181 1  and  181 3),  Laplace,  Ampere  (1806),  Poinsot 
(1806),  and  -others,  ;see  Voss,  Encykl.  der  math, 
Wiss.,  iv,  I,  pp.  6y-yS' 

P.  68,  line  25  :   "Academy." 

On  the  papers  of  Maupertuis  and  Euler  on  this 
"law  of  rest,"  see  the  Monist  for  July  191 2 
(vol.  ocxii,  pp.  416-417,  436-437,  441-444),  or  the 
reprint  in  my  book  :  The  Principle  of  Least  Action 
(Chicago  and  London,  1913),  pp.  3-4,  23-24,  28-31. 
See  also  the  paper,  referred  to  on  p.  73  of  the 
Mechanics,  by  the  Marquis  de  Courtivron,  entitled : 
"  Recherches  de  Statique  et  de  Dynamique,  ou 
Ton  donne  un  nouveau  principe  general  pour  la 
consideration  des  corps  animes  par  des  forces 
variables,  suivant  une  loi  quelconque,"  Histoire  de 
PAcademie  Roy  ale  des  Sciences.  Annee  1749.  Avec 
les  Memoires  de  Math,  et  de  Phys.  pour  la  rnenie 
Annife,  Paris,  1753,  pp.  15-27  of  the  Memoires  {on 
pp.  177-179  of  the  Histoiri  there  is  a  short  account 
of  this  memoir).  The  title  and  enunciation  of  the 
principles  were  given  in  the  Memoires  for  1748, 
published  in  1752  {cf.  Mc^ moires  for  1747,  published 
in  1752,  p.  698).  The  principle  is  that  of  all  the 
positions  which  a  system  of  bodies  animated  by  any 


70        THE  SCIENCE  OF  MECHANICS 

forces  and  connected  in  any  way  takes  successively, 
that  where  the  system  has  the  greatest  Zwz;^,  is  the 
same  as  that  in  which  it  would  have  been  necessary 
to  put  it  in  the  first  place  in  order  that  it  might  stay 
in  equilibrium. 

P.  ^6,  line  8  up:    "enunciated." 

The  actual  quotation  (from  the  second  edition  of 
the  Vorlesungen  ilber  Dynamik  in  the  Supplement- 
band  of  Jacobi's  Gesammelte  Werke,  p.  15)  is  as 
follows.  Jacobi  is  speaking  of  the  transition  from 
Lagrange's  variational  form  of  d'Alembert's  principle 
for  a  system  of  independent  masses  to  that  for  a 
system  in  which  there  are  equations  of  condition, 
and  the  displacements  of  the  co-ordinates  are  virtual. 
''The  above  extension  of  our  symbolic  equation  to 
a  system  limited  by  conditions  is,  of  course,  not 
proved,  but  only  historically  asserted.  To  say  this 
appears  necessary,  for,  although  Laplace  did  not 
prove  this  extension  in  his  M^canique  celeste  any 
more  than  I  have  done  here,  yet  this  remark  of 
Laplace's  has  been  considered  to  be  a  proof.  Poinsot 
{Journ.  de  Math.^  vol.  iii,  p.  244)  wrote  a  paper 
against  this  opinion,  and  said  very  correctly  that 
mathematicians  are  often  deceived  by  the  very  long 
ways  that  they  have  traversed,  and  sometimes  also 
by  the  very  short  ones.  They  are  deceived  if  they 
finally  come,  by  means  of  very  lengthy  calculations, 
to  an  identity,  but  hold  it  to  be  a  theorem.      An 


NOTES  ON  MACH'S   ''MECHANICS''     71 

example  of  the  other  kind  of  deception  is  given  by 
our  case.  To  prove  this  extension  is  in  no  way  my 
intention  ;  we  will  regard  it  as  a  principle  which  it 
is  not  necessary  to  prove.  This  is  the  view  of  many 
mathematicians,  in  particular  is  it  that  of  Gauss." 
And  Clebsch  added  a  note  to  say  that  that  was 
probably  a  verbal  communication  to  Jacobi. 

P.   109,  line  15  : 

It  may  here  be  mentioned  that  convenient  German 
translations  or  reprints,  as  the  case  may  be,  of  the 
fundamental  memoirs  of  Green  and  Gauss  {cf.  p.  398 
of  the  Mechanics)  on  the  theory  of  potential  are 
published  in  Ostwald's  Klassiker^  as  Nos.  61  and  2 
(edited  by  A.  von  Oettingen  and  A.  Wangerin) 
respectively. 

P.   no,  line  3  :   **  source." 
Cf,  Mechanics^  pp.  395-402. 

P.  118,  line  4:   '*  1672." 

A  convenient  German  translation  with  notes,  by 
F.  Dannemann,  made  No.  59  of  Ostw aid's  Klassiker, 
under  the  title:  Neue  ''  Magdeburgische"  Versuche 
fiber  den  leeren  Raum. 

P.   130,  line  I  :    "bodies." 

These  investigations  are  contained  in  the  dis- 
courses for  the  third  and  fourth  day  of  Galileo's 
Discorsi  e  dimostrazioni  matematiche  intorno  a  due 


72         THE  SCIENCE  OF  MECHANICS 

nuove  sctenze,  Ley  den,  1638,  of  which  a  convenient 
German  translation,  with  notes,  was  given  by  A.  J. 
von  Oettingen  in  Ostwald's  Klassiker^  No.  24.  The 
discourses  for  the  first  and  second  days  are  translated 
in  No.  II  of  the  same  collection  ;  while  No.  25  con- 
tains the  appendix  to  the  third  and  fourth  day,  and 
the  fifth  and  sixth  day.  The  title  of  an  English 
translation  is  given  on  p.  20  above. 

P.   130,  line  II  :   "following." 
Klassiker,  No.  24,  pp.  16-17. 

P.   130,  last  line  :   *'  on." 

See  Mecha7iics,  pp.  247-248. 

P.   131,  line  19:   "correct." 

Klassiker,  No.  24,  pp.  2 1  et  seqq. 

P.   132,  line  22  :   "table." 
Klassiker,  No.  24,  p.  24. 

P.   133,  line  3  : 

Cf>  Conservation  of  Energy ^  pp.  23-28. 

P.  136,  line  3  :   *'  side." 
Klassiker,  No.  24,  p.   19. 

P.   141,  line  9  up:   "of  gravity." 

As    Lagrange    does    in  his   Mecanique  {cf.^   e.g.^ 
CEuvres,  vol.   xi,  p.   239). 

P.   158,  line  2  :   "force." 

Joseph    Bertrand,    in    a    note    to    his    edition    of 


NOTES  ON  MACH'S  ''MECHANICS''     73 

Mdcanique  {cf.  CEuvres  de  Lagrange^  vol.  xi,  p.  238), 
remarked  that  a  part  of  Galileo's  Dialogo  sopra  le 
due  massivii  sistemi  del  i}io7ido  .  .  .(Florence,  17 10, 
pp.  185  et  seqq.)  seemed,  in  spite  of  a  grave  error, 
to  contain  the  germ  of  Huygens'  discovery  of  centri- 
fugal force. 

Huygens  communicated  some  theorems  on  centri- 
fugal force  to  the  Royal  Society  of  London,  in  the 
form  of  an  anagram,  in  1669,  and  thirteen  theorems 
on  centrifugal  force  were  given,  without  proof,  in 
the  fifth  part  of  the  Horologium  oscillatorium  of 
1673..  The  proofs  of  these  theorems  were  given 
in  the  Tractatus  de  vi  centrifuga^  published  after 
Huygens'  death  in  the  Opuscula  postuma,  Ley  den, 
1703.  Of  this  there  is  a  convenient  German  trans- 
lation, with  notes  by  F.  Hausdorff,  in  Ostwald's 
Klassiker,  No.  138,  pp.  35-67,  72-79.  The  Horo- 
logiu77i  is  translated  in  No.  192  of  this  collection  by 
A.  Hecksher  and  A.  von  Oettingen. 

P.   160,  line  2  : 

Descartes,  in  his  G  come  trie  of  1637,  used  equa- 
tions instead  of  proportions,  and,  though  Wallis 
introduced  the  practice  of  working  with  equations 
instead  of  proportions  in  mechanics,  in  his  treatise 
Mechanica :  sive  De  Motti,  Tractatus  Geometricus^ 
London  (parts  i  and  ii,  1670  ;  part  iii,  167 1), 
Newton  still  used  proportions  in  his  Principia  of 
1687.      It   seems  that  the  language  of   proportions 


74         THE  SCIENCE  OF  MECHANICS 

causes  less  difficulties  to  a  beginner,  because,  for 
example,  the  proportion  ''the  velocities  are  as  the 
times"  cannot  be  misunderstood  so  readily  as  the 
equation  v=^gt.  A  beginner  would  think  that  we 
were  equating  the  velocity  to  the  time  multiplied  by 
a  certain  constant,  and  most  authors  (cf.  Mechanics, 
p.  269)  encourage  this  misunderstanding  by  their 
sacrifice  of  accuracy  to  brevity.  The  truth  is,  of 
course,  that  ''z;"  does  not  stand  for  the  velocity  but 
for  the  numerical  measure,  in  terms  of  some  unit  of 
velocity  ;  and  so  on. 

P.   161,  line  4  up  {if.  p.  251)  : 

The  references  are  :  Richer,  Recueil  d' Observa- 
tions faites  en  Plusieurs  Voyages,  .  .  .  Paris, 
1693  ;  Huygens,  Discours  de  la  Cause  de  la 
Pesanteur,  1690,  p.  145.  Cf.  Todhunter,  A  History 
of  the  Mathematical  Theories  of  Attraction  and  the 
Figure  of  the  Earth,  from  the  Time  of  Newton  to  that 
of  Laplace,  London,   1873,  vol.  i,  pp.  29-30. 

P.   167,  line  13:   "imaginable." 

On  the  principle  of  similarity  with  Aristotle, 
Galileo,  Newton,  Joseph  Bertrand  (1847)  i"  especial, 
and  many  others,  see  Encykl.  der  math.  Wiss.,  iv, 
I,  pp.  23  (A.  Voss,  1901),  478-480  (P.  Stackel, 
1908). 

P.   173,  Hne  3  : 

Cf.  Conservation  of  Energy,  pp.  28-30. 


NO  TES  ON  MA  CH'S   * '  ME  CHA  NICS  "     7  5 

P.   175,  line  9  : 

Cf.  Mach,  Conservation  of  Energy,  pp.  28-30. 
Some  light  is  thrown  on  Huygens'  principle  by 
the  following  considerations.  One  of  Galileo's 
fundamental  equations  takes  the  form  {cf.  pp.  269- 
270  of  Mechanics) 

If  we  introduce  the  conception  of  mass,  we  can  say 
that  Huygens  generalised  this  into  the  form  {cf. 
p.   \  J Z  oi  Mechanics) 

and  this  again,  when  the  forces  are  not  necessarily 
constant  nor  the  paths  of  the  masses  rectilinear  {cf 
Mechanics,  pp.  276-277,  343-344,  350),  becomes 

2//  .  ds  =  ^^7?t{v^  —  "^o^), 

or,  if  a  force-function  U  exists, 

U-Uo  =  T-To, 

in  the  usual  notation.  Now,  this  is  a  frst  integral 
of  the  equations  of  motion  of  a  system  of  masses, 
and  thus,  if,  as  in  the  case  of  the  problem  of  the 
centre  of  oscillation,  there  is  only  one  degree  of 
freedom,  this  integral  gives  the  complete  solution 
of  the  problem  {cf.  my  Least  Action,  referred  to 
above,  pp.  69-76). 

P.   177,  last  line  : 

For  continuation,  see  Mechanics,  p.  331. 


^6        THE  SCIENCE  OF  MECHANICS 

P.   187,  line  7  up  : 

The  observations  of  Tycho  Brahe  enabled  his 
friend  and  pupil,  Johann  Kepler  (i  571-1630),  to 
subject  the  planetary  motions  to  a  far  more  searching 
examination  than  had  yet  been  attempted.  Kepler 
first  endeavoured  to  represent  the  planetary  orbits 
by  the  hypothesis  of  uniform  motion  in  circular 
orbits  ;  but,  in  examining  the  orbit  of  Mars,  he 
found  the  deviations  from  a  circle  too  great  to  be 
owing  to  errors  of  observation.  He  therefore  tried 
to  fit  in  his  observations  with  various  other  curves, 
and  was  led  to  the  discovery  that  Mars  revolved 
round  the  sun  in  an  elliptical  orbit,  in  one  of  the 
foci  of  which  the  sun  was  placed.  By  means  of  the 
same  observations  he  found  that  the  radius  vector 
drawn  from  the  sun  to  Mars  describes  equal  areas 
in  equal  times.  These  two  discoveries  were  extended 
to  all  the  other  planets  of  the  system  and  were 
published  at  Prague  in  1609  in  his  Nova  Astronomia 
seu  Physica  Ccelestia  tradita  Coinmentar'iis  de  Motibus 
Stellce  Martis.  In  16 19  he  published  at  Linz  his 
Harmonia  Miindi,  which  contained  his  third  great 
discovery  —  that  the  squares  of  the  periodic  times 
of  any  two  planets  in  the  system  are  to  one  another 
as  the  cubes  of  their  distances  from  the  sun. 

P.   188,  Hne  II  : 

That  the  paths  of  the  planets  were  to  be  explained 
by    a   force   constantly  deflecting  the  planets  from 


NOTES  ON  MACH'S  ''MECHANICS''     77 

the  straight  line  which  they  tend  to  describe 
uniformly  was  contemplated,  before  Newton,  by 
Wren,  Hooke,  and  Halley  {cf.  Halley's  letter  to 
Newton  of  June  29th,  1686,  printed  in  W.  W.  Rouse 
Ball's  Essay  on  Newton's  ''  Principia,"  London  and 
New  York,  1893,  P-  162).  Hooke,  indeed  {ibid., 
p.  151  ;  and  cf.  for  a  fuller  account,  a  paper  in  the 
Monist  for  July  19 13,  vol.  xxiii,  pp.  353-384), 
read  before  the  Royal  Society  in  1666  a  paper 
explaining  the  inflection  of  a  direct  motion  into 
a  curve  by  a  "supervening  attractive  principle." 
All  three  seem  to  have  been  stopped  by  the 
mathematical  difficulties  of  the  problem.  It  may 
be  noticed  that  Halley,  presumably  in  much  the 
same  manner  as  that  used  by  Newton  {cf.  Mechanics, 
p.  189),  concluded  that  the  centripetal  force  varied 
•  inversely  as  the  square  of  the  distance. 

P.   189,  line  1 1  up  :   "analysis." 

Newton  {Prittcipia,  book  i,  section  viii)  derived 
his  law  of  attraction  from  Kepler's  laws.  John 
Bernoulli  {Mem.  de  VAcad.  de  Paris,  1710,  p.  521  ; 
Opera,  vol.  i,  p.  470)  showed  conversely  that  a 
central  force  reciprocally  proportional  to  the  square 
of  the  distance  leads  to  a  Kepler's  motion  in  a 
conic  section.  Cf  P.  Stackel,  Encykl.  der  math. 
IViss.,  iv,  I,  1908,  p.  494. 

P.   189,  last  line  : 

This  achievement  of  the  imagination  was,  it  seems 


78        THE  SCIENCE  OF  MECHANICS 

(see  my  article  in  the  Monist  for  July  19 13),  also 
performed,  before  Newton,  by  Robert  Hooke. 
What  chiefly  distinguishes  Newton's  work  on  the 
theory  of  universal  gravitation  from  that  of  Hooke 
is  the  far  greater  mathematical  ability  of  Newton. 
It  was  partly  Newton's  good  luck  that,  at  the  time 
(1665-66)  when  he  began  his  scientific  career, 
everything  was  ripe  for  the  formation  of  a  mathe- 
matical method  out  of  the  infinitesimal  and  fluxional 
ideas  then  current  ;  it  is  this  formation  that  really 
seems  to  be  the  decisive  factor  in  the  |^ewtonian 
mechanics  of  the  heavens.  As  for  the  Newtonian 
principles  of  mechanics,  they  seem  to  have  grown  up 
as  a  result  of  the  Newtonian  theory  of  astronomy. 
Thus  the  conception  of  mass  as  distinguished  from 
weight  and  the  third  law  of  motion  plainly  had 
their  origin  in  extra-terrestrial  considerations.  We 
can  trace  noteworthy  approximations  to  the  New- 
tonian standpoint  with  respect  to  both  these  questions 
with  Wallis  and  Hooke  {cf.  my  article  just  cited). 

P.   191,  line  17  :   "  earth." 

Newton  had  satisfied  himself,  as  early  as  1666, 
that  the  moon  was  kept  in  her  orbit  by  a  gravita- 
tional force  towards  the  earth,  and  had  begun  to 
suspect  that  gravitation  was  a  universal  property  of 
matter  ;  but  at  that  time  he  seems  to  have  supposed 
that  masses  of  sensible  size  could  only  behave  to 
one  another    approximately  as    attracting  points  at 


NOTES  ON  MACtrS  ''MECHANICS''     79 

very  great  distances  from  one  another.      But  it  was 
only  in   1685  {cf.  W.  W.  Rouse  Ball,  An  Essay  on 
Newton's  "■  Pn'ndpia,"  London  and  New  York,  1893, 
pp.  116,  157)  that  Newton  discovered  that  a  spherical 
mass  attracts  external  masses  as  if  the  whole  mass 
were  collected  at  the  centre  (this  forms  Proposition 
7 1  of  section  xii  of  the  first  book  of  the  Principid), 
*'  No  sooner,"  said  J.   W.    L.  Glaisher  at  the  com- 
memoration (1887)  of  the  bicentenary  of  the  publica- 
tion of  the  Principia  (see  Ball,  op.  cit.^  p.  61),  "had 
Newton  proved  this  superb  theorem — and  we  know 
from  his  own  words  that  he  had  no  expectation  of 
so  beautiful  a  result  till  it  emerged  from  his  mathe- 
matical investigation — than  all  the  mechanism  of  the 
universe  at  once  lay  spread  before  him.      When  he 
discovered  the    theorems    that    form  the  first  three 
sections  of  Book  I  [of  the  Principia\  when  he  gave 
them  in  his  lectures  of  1684,  he  was  unaware  that 
the  sun  and  earth  exerted  their  attractions  as  if  they 
were    but    points.      How  different    must   these  pro- 
positions have    seemed    to  Newton's  eyes  when   he 
realised  that  these  results,  which  he  had  believed  to 
be  only  approximately  true  when  applied  to  the  solar 
system,  were  really  exact !     Hitherto  they  had  been 
true  only  in  so  far  as  he  could  regard  the  sun  as  a 
point  compared  to  the  distance  of  the  planets,  or  the 
earth    as  a   point  compared    to  the  distance  of  the 
moon, — a  distance  amounting  to   only  about  sixty 
times  the  earth's  radius — but  now  they  were  mathe- 


8o         THE  SCIENCE  OF  MECHANICS 

matically  true,  excepting  only  for  the  slight  deviation 
from  a  perfectly  spherical  form  of  the  sun,  earth,  and 
planets.  We  can  imagine  the  effect  of  this  sudden 
transition  from  approximation  to  exactitude  in 
stimulating  Newton's  mind  to  still  greater  efforts. 
It  was  now  in  his  power  to  apply  mathematical 
analysis  with  absolute  precision  to  the  actual 
problems  of  astronomy." 

P.   192,  line  7  up  : 

The  Principia  is  reprinted  in  the  second  volume 
of  the    only  complete    edition  of   Newton's  works, 
which  was  published  in  five  volumes  at  London  in 
1779-85,    under    the    title  :    Isaaci   Newtoni    Opera 
qucB  exstant  omnia  Comvietitariis  illustrabat  Samuel 
Horsley.      Further   details    about    the    editions    and 
translations  of  this  and  other  works  of  Newton  are 
to  be  found  in  G.  J.  Gray's  book  :   A  Bibliography  of 
the    Works    of  Sir   Isaac   Newton^   together  with  a 
List  of  Books    illustrating  his    Works,   Cambridge, 
1907.      The  best-known  English  translation  of  the 
Principia   is  by  Andrew  Motte  {The  Mathematical 
Principles  of  Natural  Philosophy,  2  volumes,  London, 
1729  ;  American  editions  in  one  volume,  New  York, 
1848    and     1850).      This    translation    includes    the 
preface  which  Roger  Cotes  prefixed   to  the  second 
edition  of  17 13  of  the  Principia,  which  was  edited 
by  him  in  Latin.      A  third  edition  was  edited    by 
Henry  Pemberton  in   1726. 


NOTES  ON  MACH'S  ''MECHANICS''     8i 

F.  200,  line  16  : 

It  is  remarkable  that  this  passage  in  the  Principia 
(Scholium  to  the  Laws  of  Motion)  was  not  referred 
to  by  Mach  in  his  tract  on  the  Conservation  of 
Energy  as  showing  that  the  principle  of  the  excluded 
perpetuum  mobile  lies  at  the  bottom  of  our  instinctive 
perception  of  the  truth  of  the  third  law.  That  this 
is  so  was  recognised  by  J.  B.  Stallo  (The  Concepts 
and  Theories  of  Modern  Physics^  fourth  edition, 
London,  1900 ;  cf.  the  references  in  my  notes  to 
Mach's  Conservation  of  Energy,  pp.  98,  99,  lOi), 
whose"^  views  on  the  part  played  in  science  of  all 
ages  by  the  principle  of  energy  very  closely  resemble 
those  of  Mach.  Otherwise  the  third  law  seems  by 
no  means  to  be  the  plain  expression  of  an  instinctive 
perception.  By  Hertz's  evidence  {cf  Mechanics, 
pp.  549-550)  and  the  experience  of  some  of  us  when 
being  taught  mechanics,  we  know  that  the  point  is 
often  not  grasped,  and  there  is,  in  the  traditional 
Newtonian  form,  no  appeal  to  what  instinctive 
knowledge  we  may  possess.  Cf  §  xiv  of  my 
article  in  the  Monist  for  October   19 14  (vol.  xxiv, 

pp.  553-555). 

P.  201,  line  18  : 

An  account  of  Newton's  achievements  is  also  given 
by  E.  Diihring  on  pp.  172-21 1  of  his  Kritische 
Geschichte  der  allgemeinen  Principien  der  Mechanik, 
3rd  ed. ,  Leipsic,  1887.      Diihring  does  not,  however, 


82         THE  SCIENCE  OF  MECHANICS 

criticise    Newton's    conception    of    mass.       Cf.    my 
article  in  the  Monist  for  April  and  October  19 14. 

P.  216,  line  8  up  : 

Rosenberger  {Isaac  Newton  und  seine  pJiysika- 
lischen  Principieny  Leipsic,  1895,  p.  173)  has 
practically  remarked  that  this  is  what  Newton  him- 
self did.  "From  what  Newton  says  further  on,  it 
appears  that  he  supposed  that  all  of  the  smallest 
particles  of  matter  are  equally  dense  and  of  the 
same  size,  and  put  the  density  proportional  to  the 
number  of  these  particles  in  a  given  space."  Cf, 
my  article  in  the  Mo?iist  for  October  1 9 14. 

P.  238,  line  5  : 

On   the  law    of   inertia    see  also  Conservation  of 
Energy^  pp.  75-80. 

P.  244,  line  II  :   "convention." 

Cf.  Yoss's  {Encykl.  der  math.  IViss.,  iv,  i,  1901, 
pp.  49-50)  fundamental  propositions  of  dynamics. 
The  account,  with  many  references  (idid.,  pp.  50- 
56),  of  critical  researches  on  the  independence  of 
Newton's  axioms,  the  concept  of  mass,  the  principle 
of  inertia,  the  conception  of  force,  and  the  law  of 
action  and  reaction  should  also  be  consulted. 

P.  278,  Hne  9,  "  Gauss.'' 

Gauss's  paper  on  the  reduction  of  the  intensity 
of  the  force  of  terrestrial  magnetism  to  absolute 
measure  ("  Intensitas  vis  magneticae  ad  mensuram 


NOTES  ON  MACITS   ''MECHANICS''     83 

absolutam  revocata,"  Gott.  AbJi.,  1832;  Werke^ 
vol.  V,  p.  81)  was  originally  printed  in  Latin,  and 
a  convenient  German  edition,  edited  by  E.  Dorn, 
is  published  in  No.  53  of  Ostwahfs  Klassiker. 

P.  278,  line  17  : 

On  the  looseness  of  phrase  according  to  which  it 
is  customary,  in  most  treatises  on  mechanics  and 
geometry,  to  talk  about  /,  ^,  and  v  simply  as  the 
time,  the  distance,  or  the  velocity,  instead  of  the 
numerical  measures  of  these  quantities,  see  the  above 
note  to  p.   160. 

P.  288,  line  9  up  : 

The  law  of  the  conservation  of  momentum  was 
given  by  Newton  in  the  third  Corollary  to  his  Laws 
of  Motion,  and  that  of  the  conservation  of  the 
centre  of  gravity  in  the  fourth  Corollary.  Both 
are  translated,  together  with  the  papers  of  Daniel 
Bernoulli  and  d'Arcy  on  the  law  of  the  conservation 
of  areas  (see  p.  293  of  Mecltanics,  and  my  note  on  it 
below),  and  that  of  Daniel  Bernoulli,  of  1748,  on  the 
principle  oi  vis  viva  (see  pp.  343,  348  oi  Mechanics^ 
and  my  note  to  p.  343  below),  in  No.  191  of  OstwalcVs 
Klassiker  {Abhandlungen  iiber  jene  Prinzipien  der 
MecJianik^  die  Integrate  der  dynamischen  Differential- 
gleichungen  liefern,  von  Newton  {168"/),  Daniel 
Bernoulli  {174.3,  ^74^)  ^^^  d'Arcy  {1747))^  edited  by 
Philip  E.  B.  Jourdain. 


84        THE  SCIENCE  OF  MECHANICS 

P.  293,  last  line  : 

The  references  are:  Daniel  Bernoulli,  '*  Nouveau 
probleme  de  mecanique  resolu  par  .  .  .,"  Hist,  de 
PAcad.  de  Berlin,  vol.  i,  1745  (published  1746), 
pp.  54-70;  Euler,  "  De  Motu  Corporum  in  super- 
ficiebus  mobilibus,"  L.  Euleri  Opuscula  varii 
aygumetiti,  vol.  i,  Berlin,  1756,  pp.  1-136  {cf. 
Diihring,  op.  cit.,  pp.  285-286);  Patrick  d'Arcy, 
"Probleme  de  Dynamique,"  Hist,  de  TAcad.  Roy. 
des  Sci.,  77^7  (Paris,  1752)  ;  Alemoires,  pp.  344-361 
(this  consists  of  three  memoirs  read  in  1743,  1746, 
and  1747  respectively  ;  it  is  the  second  that  contains 
the  statement  of  the  principle  in  question).  On  all 
these  papers,  see  Ostwalds  Klassiker,  No.   191. 

On  d'Arcy's  later  statement  (1749)  of  his  principle 
of  areas  in  a  form  which  seemed  to  him  to  be 
preferable  to  that  of  Maupertuis'  principle  of  least 
action,  while  answering  the  same  purpose  ;  and  on 
the  discussion  arising  from  this  between  d'Arcy, 
Maupertuis,  and  Louis  Bertrand,  see  Monist,  July 
191 2,  vol.  xxii,  pp.  445-456,  or  my  Least  Action, 
pp.  32-43. 
P.  313,  line  15:   "theorems." 

Wren    admitted    that    he    could    not    prove    his 
theorems. 

P.  313,  line  17  : 

These  experiments  of  Wallis,  Wren,  and  Huygens 
are  referred  in  P'elix  Hausdorff' s  notes  to  the  German 


NOTES  ON  MACH'S  ''MECHANICS''     85 

translation  in  No.  138  of  Ostwalds  Klassiker^  of 
Huygens'  posthumous  treatises  referred  to  on  p.  314 
(see  the  next  note)  of  Mechanics.  Wallis's  treatise, 
referred  to  on  p.  313  of  Mechanics^  is  entitled 
Mechanica :  sive  De  Motu,  Tract atus  Geometricus, 
and  appeared  at  London  in  three  parts  :  Parts  I  and 
II  in  1670,  and  Part  III,  which  contains  the  section 
'*  De  Percussione,"  appeared  in  1671.  Of  interest  in 
this  connection  are  the  following  extracts  : — p.  4  : 
"■  Per  Pondus  intelligo  gravitatis  mensuram  "  ;  p.  5  : 
''  Pondus  sic  intellectum,  aut  Gravitatis  etiam,  prout 
vel  in  Movente  vel  in  Mobili,  considerato  ;  ita  vel 
ad  Movendi,  vel  ad  Resistendi  vim  partinebit : 
Adeoque  nunc  ad  Momentum,  nunc  ad  Impedi- 
mentum  referetur." 

P.  314,  line  7  :   ''  1703." 

Huygens  first  pubHshed  his  laws  of  impact  in  a 
paper  called  "  The  Laws  of  Motion  on  the  Collision 
of  Bodies,"  in  the  Philosophical  Transactions  for 
1669,  and  in  *' Regies  du  mouvement  dans  la 
rencontre  des  corps,"  in  the  Journal  des  Savants. 
But,  a  year  earlier,  Huygens  spoke  on  these  laws 
in  the  Paris  Academy  ;  and  their  discovery  must 
date  much  further  back,  for,  in  a  letter  to  Claude 
Mylon  of  July  6th,  1656,  Huygens  mentioned  the 
increase  of  the  action  of  impact  by  the  interposition 
of  an  intermediate  body.  The  extended  derivation 
of  the  laws — the  results  of  1669  being  given  without 


86         THE  SCIENCE  OF  MECHANICS 

proof — was  given  in  the  Tractatus  de  motu  corporum 
expercussione^  which  appeared  in  Huygens'  Opnscula 
postuma,  edited  by  Burcherus  de  Voider  and  Bern- 
hardus  Fullenius  (Lugduni  Batavorum,  1703).  A 
German  translation  of  this  Tractatus,  with  notes 
by  F.  Hausdorff,  was  given  in  1903  in  Ostwald's 
Klassiker,  No.   138,  pp.  3-34,  63-72. 

P.  330,  last  line  : 

Further  references  on  the   subject  of  impulses  are 
to  be  found  in  Voss's  article  in  the  Encykl.  der  math, 
Wzss.,  iv,   I,  190 1,  pp.  56-58,  87-88. 

P.  336,  line  21  : 

The  greater  part  of  d'Alembert's  Traits  de  dyn- 
aniique  was  translated  into  German  and  annotated 
by  Arthur  Korn  in  No.   106  of  Ostwald's  Klassiker, 

P.  343,  line  12  up  {cf.  p.  348,  line  4  up)  : 

Daniel  Bernoulli's  paper,  "  Remarques  sur  le 
principe  de  la  conservation  des  forces  vives  pris 
dans  un  sens  general,"  was  published  in  1750  in  the 
Hist,  de  VAcad.  de  Berlin  for  1748,  pp.  356-364 
(among  the  Mcmoires  de  la  Classe  de  philosophie 
speculative).  Jacobi  {op.  cit.,  pp.  9-10)  remarked 
that  Daniel  Bernoulli  first  noticed  that  one  and  the 
same  U  could  serve  for  all  the  masses  in  the  problem. 
In  fact  it  is  evident  that  partial  differentiation  with 
respect  to  the  co-ordinate  x  only  affects  x,  and  con- 
sequently only  those  co-ordinates  which  multiply  x 


NOTES  ON  MACH'S  ''MECHANICS''     Zj 

enter  the  result.  Bernoulli  went,  according  to 
Jacobi,  beyond  Euler,  and  his  point  of  view  was 
developed  by  Lagrange. 

P.  350,  line  2  up  of  text  :   "  233." 

"  Uber  ein  neues  allgemeines  Grundgesetz  der 
M^chdimk,''  Journ.  filr  Math. ,wo\.  iv,  1829;  Werke, 
vol.  V,  pp.  23-28  ;  Ostwald's  Klassiker^  No.  167, 
pp.  27-30. 

P.  352,  line  10:   '' D'Alembert." 

Gauss's  principle  is  more  general  than  d'Alembert's, 
in  that  it  embraces  cases  where  the  conditions  can 
only  be  expressed  by  equalities  {cf,  Voss,  Encykl. 
der  math.  Wzss.,  iv,  i,  1901,  p.  S6,  note  229). 
Where  the  conditions  are,  as  is  usually  the  case, 
expressible  by  equations,  Gauss's  principle  can,  of 
course,  be  deduced  from  d'Alembert's.  Cf.  also 
Ostwald's  Klassiker,  No.  167,  pp.  64-65,  and,  for 
another  advantage  of  Gauss's  principle,  pp.  47-4^. 

P.  361,  line  9  up  :   '*  principle." 

On  the  analytical  expression  of  Gauss's  principle, 
see  my  notes  in  Ostwald's  Klassiker,  No.  167, 
pp.  47,  60-67.  Here  arises  the  interesting  question, 
not  dealt  with  by  Mach,  as  to  whether  the  Gaussian 
process  of  variation  affects  the  co-ordinates  and 
velocities,  or  only  the  accelerations.  Mach's  remark 
on  p.  362,  lines  5-8,  of  his  Mechanics  had  been 
made    by    J.    W.    Gibbs    (''On    the    Fundamental 


88         THE  SCIENCE  OF  MECHANICS 

Formulse    of  Dynamics,"   Amer.  Jourti.    of  Math.^ 
vol.  ii,  1879,  pp.  49-64). 

P.  364,  line  7  up  : 

The  early  history  of  the  principle  of  least  action 
in  Maupertuis's  hands  is  described  in  detail  in  my 
paper  in  the  Monist  for  July  19 12,  or  my  Least 
Action^  pp.  1-46.  The  first  account  of  Maupertuis's 
principles  was  given  in  a  memoir  read  to  the  French 
Academy  in  1744, and  entitled  "Accord  de  differentes 
Loix  de  la  Nature  qui  avoient  jusqu'ici  paru  incom- 
patibles "  {Histoire  de  VAcadimie^  Annee  1744. 
(Paris,  1748),  pp.  417-426),  whereas  Maupertuis's 
memoir  "  Les  Loix  du  mouvement  et  du  Repos 
deduites  d'un  Principe  Metaphysique  "  was  printed 
in  the  Histoire  de  t Academie  de  Berlin  for  1746, 
pp.  267-294.  Mach's  date  of  1747  is  thus  a  mistake : 
see  the  Monist,  April  19 1 2,  vol.  xxii,  p.  285,  or 
my  Least  Action ,  p.  47. 

P.  367,  line  13  :   ''sense." 

According  to  P.  Stackel  {Encykl.  der  math.  Wiss. , 
iv,  I,  1908,  p.  491,  note  125),  Mach  wrongly 
supposed  that  Maupertuis  worked  on  the  basis  of 
the  undulatory  theory  of  light,  whereas  he  really 
adopted  the  emission-theory,  like  a  good  Newtonian; 
and  hence  Mach  mistakenly  found  a  contradiction 
in   Maupertuis's  treatment. 

Further,  Maupertuis's  principle  does  state  that 
whdX  fv ,  ds    reduces    to    in    this    case    is    to    be   a 


NOTES  ON  MACH'S  ''MECHANICS''     89 

minimum,  and  this  was  contested  by  Mach.      How- 
ever, cf.  pp.  375-376  of  his  Mechanics. 

On  the  subject  of  the  principle  of  least  action 
and  the  case  of  the  motion  of  light,  see  Monist, 
April  1912,  vol.  xxii,  pp.  285-288,  and  July  1912, 
vol.  xxii,  pp.  417-419 ;  or  my  Least  Action, 
pp.   47-50  and  4-6  respectively. 

P.  368,  line  16  up  :   ^^Euler's." 

Methodus  inveniendi  lineas  curvas  maximi  mini- 
mive  prop  vie  tat  e  gaudentes  :  sive  solutio  problematis 
isoperinietrici  latissimo  sense  accepti  ;  Lausanne  and 
Geneva,  1744 ;  second  appendix  :  De  motu  pro- 
jectoruin  in  medio  non  resistente.  In  the  German 
translation  of  a  part  of  the  Methodus  in  Ostwald's 
Klassiker,  No.  46  (among  the  classical  works  on 
the  calculus  of  variations),  this  appendix  does  not 
appear. 

The  Methodus  was  published  in  the  autumn  of 
1744,  some  months  after  Maupertuis's  first  paper  on 
least  action  was  presented  to  the  French  Academy  ; 
but,  as  A.  Mayer  {Geschichte  des  Princips  der 
kleinsten  Action,  Akademische  Antrittsvorlesung, 
Leipsic,  1877)  pointed  out,  Euler's  discovery  was 
made  under  the  stimulus  of  the  Bernoullis,  and 
independently  of  Maupertuis ;  but  tliat  later  on 
Euler's  own  tendency  towards  metaphysical  specula- 
tion combined  with  the  influence  of  Maupertuis  to 
make  Euler   treat   his  principle    in  a  more  general 


90        THE  SCIENCE  OF  MECHANICS 

and  a  priori^  and  less  precise  way.  Cf.  my  notes  in 
Ostwald's  Klassiker,  No.  167,  pp.  31-37;  and  my 
papers  in  the  Monist  for  April  (pp.  288-289)  and 
July  (pp.  429-445,  456-459)  1912;  or  my  Least 
Action^  pp.  50-51  and  16-32,  43-46  respectively. 

P.  371,  line  8  :  ''holds." 

This  is  not  correct  :  Lagrange,  in  the  memoir 
quoted  below,  in  which  he  generalised  (see  Mechanics^ 
p.  380)  Euler's  theorem,  drew  attention  to  the  fact 
that  the  principle  of  least  action  does  not  depend  for 
its  validity  on  the  principle  of  vis  viva,  which  only 
follows  from  the  equations  of  mechanics  under  the 
special  condition  (see  Mechanics,  p.  478)  that  the 
connections  do  not  depend  on  the  time.  See  my 
papers  in  the  Monist  for  April  (pp.  289-292)  and 
July  (pp.  456-457)  1912;  or  my  Least  Action, 
pp.  51-54  and  43-44  respectively. 

The  title  of  Lagrange's  memoir  is  "Application 
de  la  methode  exposee  dans  le  memoire  precedent  a 
la  solution  de  differents  problemes  de  dynamique," 
Miscellanea  Taurine nsia  {or  1760  and  1761  [published 
1762],  vol.  ii,  pp.  196-298  ;  CEuvres,  vol.  i,  pp.  365- 
468.  This  memoir  immediately  followed  Lagrange's 
first  fundamental  memoir  (see  Mechanics,  p.  436)  on 
the  calculus  of  variations:  "  Essai  d'une  nouvelle 
methode  pour  determiner  les  maxima  et  les  minima 
des  formules  integrales  indefinies,"  Misc.  Taur., 
1760   and    1 76 1,    vol.    ii,    pp.     173-195  ;    CEuvres, 


NOTES  ON  MACH'S  ''MECHANICS''     91 

vol,  i,  pp.    335-362  ;   OstwalcTs  Klassiker,    No.   47, 
PP-  3-30. 

P.  371,  line  8  :   '' Jacobi." 

In  his  "  Vorlesungen  liber  Dynamik,"  2nd  ed.,  in 
Werke,  Supplementband,  Berlin,  1884,  pp.  45-49; 
Ostwald's  Klassiker,  No.  167,  pp.  18-22,  58. 
Jacobi's  sixth  lecture  and  a  part  of  the  seventh, 
which  refer  to  his  (narrower  than  Lagrange's,  as  we 
know  now)  formulation  of  the  principle  of  least 
action,  were  reprinted  in  Ostwald's  Klassiker,  No. 
167,  pp.   16-26. 

The  question  of  the  relative  generality  of  Euler's, 
Lagrange's,  Hamilton's,  and  Jacobi's  principles  is 
discussed  in  my  paper  in  the  Monist  for  April 
19 1 2,  vol.  xxii,  pp.  290-296,  or  my  Least  Action^ 
pp.  51-58. 

P.  372,  line  10  :   **  stead." 

On  these  analogies,  cf.   Mechanics,  pp.  425-427  ; 
and    P.    Stackel,    Encykl.    der  math,    Wiss.,   iv,    i, 
1908,  pp.  489-493. 
P.  380,  line  12  up  : 

This  was  first  done  in  the  Miscella?iea  Taurinensia 
for  1760  and  1761  ;  see  the  Monist  for  April 
19 1 2,   vol.  xxii,  pp.  289-291  ;  or  my  Least  Action ^ 

PP-  51-53. 

P.  381,  line  8  : 

On   the  opinions   of    Michel  Ostrogradski,    Adolf 


92         THE  SCIENCE  OF  MECHANICS 

Mayer  (1877),  Helmholtz,  and  R6thy,  that  Hamil- 
ton's principle  is  a  form  of  the  principle  of  least 
action,  and  the  clear  distinction  between  these  two 
principles  in  the  work  of  Adolf  Mayer  (1886)  and 
Otto  Holder,  see  the  Monist  for  April  19 12, 
vol.  xxii,  pp.  294-296,  301-303  ;  or  my  Least 
Action,  pp.   55-58,  63-65. 

P.  390,  note  : 

This  memoir  of  Gauss's  of  1829  is  translated  into 
German  by  Rudolf  H.  Weber,  and  annotated  by 
H.  Weber,  in  No.   135  of  OstwalcPs  Klassiker. 

P.  395  : 

The  works  cited  on  this  page  are  :  Newton, 
P7'incipia,  book  iii,  prop.  19;  Huygens,  *'Dis- 
sertatio  de  causa  gravitatis "  (first  published  in 
French  in  1690),  Opera  post huma,  vol.  ii,  p.  116; 
Bouguer,  "Comparison  des  deux  Loix  que  la  Terre 
et  les  autres  Planetes  doivent  observer  dans  la  figure 
que  la  pesanteur  leur  fait  prendre,"  Mem.  de  I' Acad, 
des  Set.  de  Paris,  1734,  pp.  21-40;  Clairaut, 
Theorie  de  la  Figure  de  la  Terre,  tiree  des  Principes 
de  VHydrostatique,  Paris,  1743  (German  translation 
by  A.  von  Oettingen,  annotated  by  Philip  E.  B. 
Jourdain,  in  No.   189  of  Ostwalds  Klassiker.) 

A  very  conscientious  and  detailed  report  on  the 
work  of  many  of  Clairaut's  predecessors  and  followers 
is  given  in  the  two  volumes  of  Isaac  Todhunter's 
work  :    A  History  of  tJu  Mathematical  Theories  of 


NOTES  ON  MACH'S   ''MECHANICS"     93 

Attraction    and   the  Figure   of  the  Earthy  from  the 
Time  of  Newton  to  that  of  Laplace  (London,  1873). 

P.  397,  line  14  : 

The  conception  of  the  pressure  (/)  at  any  point 
of  a  fluid  was  introduced  by  Euler  in  his  memoir  in 
the  Histoire  de  I' Acad,  de  Berlin,  1755,  pp.  217- 
273,  and,  according  to  Todhunter  {op.  cit.,  vol.  i, 
pp.  26,  193),  this  introduction  is  the  most  important 
progress  made  in  hydrostatics  since  Clairaut. 

P.  398,  line  19:   **  Gauss." 

Convenient  German  editions,  with  notes  by 
A.  J.  von  Oettingen  and  A.  Wangerin,  of  the 
fundamental  works  on  the  theory  of  the  potential 
of  Green  (1828)  and  Gauss  (1840),  were  published  in 
Nos.  61  and  2,  respectively,  of  Ostwald's  Klassiker. 

P.  420,  last  line  : 

Other  authors  on  hydrodynamics  are  :  d'Alembert 
(the  end  of  his  Traitc  de  Dynamique  of  1743  ;  his 
Traite  des  Fhcides,  Paris,  1744;  and  his  Essai 
cTune  nouvelle  theorie  sur  la  resistance  des  Fluides, 
Paris,  1752)  and  Euler  {Hist,  de  ^Acad.  de  Berlin, 

1755). 

P.  425,  line  15  : 

At  this  point  we  may  refer  to  the  researches  of 
Maupertuis  on  the  principle  of  least  action  in  the 
case  of  the  motion  of  light  :  see  the  above  note  to 
p.  367  of  Mechanics. 


94         THE  SCIENCE  OF  MECHANICS 

P.  425,  line  27  : 

The  original  proposal  and  solution  of  this  problem 
of  John  Bernoulli's  are  given  in  a  German  translation 
by  P.  Stackel,  in  No.  46  of  Ostwald's  Klassiker. 

P.  426,  line  7  :   "light." 

On  the  analogies  between  the  motion  of  masses, 
the  equilibrium  of  strings,  and  the  motion  of  light, 
see  Mechanics^  pp.  372-380. 

P.  437,  line  17  :   '*  say." 

This  supposition  does  not  seem  to  be  correct. 
For  Lagrange  {cf.  CEuvres,  vol.  i,  pp.  337,  345  ; 
Ostwald's  Klassiker^  No.  47,  pp.  5,  13)  expressly 
"varied"  the  independent  variable,  and,  partly  on 
this  ground,  held  his  method  to  be  more  general 
than  Euler's.  Thus,  if  he  had  expressed  the  action- 
integral  in  the  form  /2T .  dt,  he  would  have  made 
the  /  to  be  affected  by  the  (5.  In  Lagrange's  de- 
velopment of  the  principle  of  least  action,  the 
question  as  to  whether  t  should  be  varied  or  not  did 
not  come  up,  as  ^v  was  at  once  eliminated.  But  the 
question  is  one  of  great  interest,  and  gave  rise  to 
many  important  works  of  Rodrigues,  Jacobi,  Ostro- 
gradski,  Mayer,  and  others :  see  Ostwald's  Klassiker^ 
No.  167,  especially  pp.  50-51,  56-58;  and  my 
paper  in  the  Monist  for  April  19 1 2,  vol.  xxii, 
pp.  292-295  ;  or  T[vy  Least  Action,  pp.  55-58  (where, 
too,   the  conceptions  of  the  nature  of  a  variation  of 


NOTES  ON  MACH'S  ''MECHANICS"     95 

Euler,  Lagrange,  Lacroix,  Jacobi,  Strauch,  M.  Ohm, 
Cauchy,  and  Stegmann  are  dealt  with). 

If  /  is  to  be  varied,  we  must  regard  it,  according 
to  the  conception  of  a  * '  variation "  derived  from 
Jellett,  in  the  work  cited  in  Mechanics^  p.  437,  as 
a  function  of  another  variable,  0,  so  that  ^^  =  0,  but 
St  is  not  zero  in  general.  This  was  done  explicitly 
by  Helmholtz  ("  Zur  Geschichte  des  Princips  der 
kleinsten  Aktion,"  Sitzungsber.  der  Berliner  Akad.^ 
Sitzung  vom  10.  Marz  1887,  pp.  225-236;  Wissen- 
schafiliche  Abhandlungen,  vol.  iii,  pp.  249-263). 

P.  437,  line  7  up  of  text :   "  form." 

Jellett  (op.  cit.y  p.  i)  defines  this  as  **the  nature 
of  the  relation  subsisting  between  the  dependent 
variable  and  the  independent  variables." 

P.  438,  line  15  :   ''required." 

Jellett  {op.  cit.,  p.  2)  denoted  this  function  of  a 
function  by  F .  0,  and  assumed  {ibid. ,  p.  4)  that 
F.  (j)  +  F01  =  F(0  -f  0i).      In  the  calculus  of  variations, 

F  is  d  or  L  and  the  distributive  law  is  verified  for 

them.      Cf.  ibid.^    pp.   11,  355. 

P.  440,  line  10  : 

A  mere  plus  is  used  in  the  expression  for  DU, 
because  (Jellet,  op.  cit.^  p.  5)  the  theorem  is  perfectly 
analogous  to  the  principle  of  the  superposition  of 
small  motions  in  mechanics,  and  may  be  proved  in 
a  similar  manner. 


96        THE  SCIENCE  OF  MECHANICS 

P.  445,  line  13  :  "function." 

Jellett  {op.  cit.,  pp.  31 1-3 13)  drew  a  distinction 
between  a  mechanical  variation,  which  is  not  a  varia- 
tion but  a  displacement,  and  a  geometrical  or  mathe- 
matical one.  In  the  former,  "  0  no  longer  denotes 
the  increment  which  is  produced  by  a  change  in 
position,  by  the  motion  of  a  particle  from  one  point  of 
space  to  another."  The  rules  governing  mechanical 
variations  are  generally  the  same  as  the  rules  in  the 
calculus  of  variations. 

P.  445,  last  line  : 

This  remark  is  not  quite  correct.  At  the  be- 
ginning of  his  career  (1759),  Lagrange  announced 
his  intention  of  deriving  the  whole  of  mechanics 
from  the  principle  of  the  least  quantity  of  action. 
He  fulfilled  this  promise  in  a  long  memoir  of  1760 
and  1 76 1,  and  it  was  only  in  1764  that  the  equations 
of  mechanics  appeared  in  a  form  which  was  not  the 
equating  to  zero  of  the  variation  of  an  integral  (see 
Mo?iisty  April  191 2,  vol.  xxii,  pp.  289-293  ;  or  my 
Least  Action^  pp.  51-54).  From  after  this  early 
work  of  Lagrange  until  the  time  of  Gauss,  no  pre- 
dilection for  expressing  mechanical  principles  in  a 
maximal  or  minimal  form  appeared,  and  the  attempt 
of  Gauss  was  hardly  in  a  direction  that  would  be 
approved  of  by  Lagrange,  who,  as  time  went  on, 
became  very  ''anti-metaphysical"  {cf.  Mechanics, 
p.  457)- 


NOTES  ON  MAC  ITS   ''MECHANICS''     97 

Mach  here  only  considers  the  calculus  of  variations 
for  functions  of  one  variable.  The  case  of  many 
variables  is  important  in  mechanics,  and  one  of 
Lagrange's  great  advances  was  to  consider  this  case. 
Jellett  {op,  cit.,  pp.  18-27,  107-111)  dealt  with 
many  variables  which  may  be  connected  by  equa- 
tions of  condition.  If  the  functions  j^*,  ^,  are  inde- 
pendent of  each  other,  their  variations  will  also  be 
independent  and  arbitrary,  and  the  coefficients  of 
^y  and  ^z  under  the  integral  sign  are  each  to  be 
equated  to  zero. 

Jellett  dealt  with  Lagrange's  method  of  multi- 
pliers (see  Mechanics^  pp.  471-472)  on  pp.  20-23, 
1 15-134,  of  his  above  quoted  work,  with  non- 
integrable,   in  general,   equations  of  condition. 

P.  454,  last  line  : 

Mach  appears  to  give  Voltaire's  version  of  some 
of  the  things  dealt  with  by  Maupertuis ;  but  Mauper- 
tuis  does  not  seem,  by  his  published  writings,  to 
have  been  nearly  so  ridiculous  a  person  as  Voltaire, 
for  personal  reasons,  tried  to  make  him  appear  to 
be  ;  see  Monist,  July  191 2,  vol.  xxii,  pp.  427-428  ; 
or  my  Least  Action^  pp.   14-15. 

P.  466,  line  8  up:   "perspicuity." 

Cf.  Th.  Korner,  "  Der  Begriff  des  materiellen 
Punktes  in  der  Mechanik  der  18.  Jahrhunderts," 
Bibl.  Math.  (3),  vol.  v,  1904,  p.  15.  Euler 
{Meclianica    sive    motus   scientia  analytice    exposita^ 

7 


98        THE  SCIENCE  OF  MECHANICS 

two  vols.,  St  Petersburg,  1736)  and  d'Alembert 
(Traitc  de  dynamique,  Paris,  1743  ;  annotated 
German  translation  by  A.  Korn  of  most  of  it  in 
Ostwalds  Klassiker,  No.  106)  everywhere  used 
''natural  co-ordinates";  and  the  methodical  intro- 
duction here  of  Cartesian  co-ordinates  is  due  to 
Maclaurin  {A  Complete  System  of  Fluxiofis,  Edin- 
burgh, 1742,  arts.  465,  469,  884).  Cf.  Mechanics, 
p.  466. 

P.  466,  line  5  up:   "  1788." 

The  various  editions  of  Lagrange's  Mhanique 
are  as  follows  :  (i)  M^canique  analitique ,  Paris, 
1788,  in  one  volume  ;  (2)  Micanique  analytique, 
Paris,  vol.  i,  181 1  ;  vol.  ii  (posthumous),  1815  ;  (3) 
Mecanique  analytiqiie,  2  vols.  Paris,  1853  and  1855, 
with  notes  by  Joseph  Bertfand  ;  (4)  like  the  third 
edition,  but  with  some  additional  notes  by  Gaston 
Darboux,  in  CEiivi'es  de  Lagrange,  vols,  xi  and  xii, 
Paris,   1888  and  1889. 

D'Alembert's  principle,  in  combination  with  the 
principle  of  virtual  displacements,  appeared  in  varia- 
tional form  {cf.  Mechanics,  pp.  342-343,  468)  for 
the  first  time  in  a  prize  essay  of  Lagrange's  of  1764 
on  the  libration  of  the  moon  {CEuvres,  vol.  vi, 
pp.  5-61)  ;  and  then,  more  fully,  in  a  memoir  of  1780 
{CEuvres J  vol.  v,  pp.  5-122)  on  the  same  subject. 

It  is  well  known  that  Lagrange  founded  the 
calculus  of  the  differential  quotients  of  functions  on 


NOTES  ON  MACH'S  ''MECHANICS''     99 

the  seeking  of  the  terms  of  the  development  of  these 
functions  by  Taylor's  series,  and  thus  avoided  the 
difficulties  connected  with  infinitesimals.  In  1797, 
nine  years  after  the  publication  of  the  M^canique, 
he  had  published  a  systematic  exposition  of  this 
theory, — the  Thiorie  des  fonctions  analytiques — with 
applications  to  mechanics.  However,  in  the  pre- 
face to  the  second  edition  (181 1)  of  the  Mecanique^ 
Lagrange  {cf.  CEuvres^  vol.  xi,  p.  xiv)  remarked  : 
''  I  have  kept  the  ordinary  notation  of  the  differential 
calculus,  because  it  corresponds  to  the  system  of 
infinitesimals  adopted  in  this  treatise.  When  we 
have  well  conceived  the  spirit  of  this  system,  and 
when  we  have  convinced  ourselves  of  the  exactitude 
of  its  results  by  the  geometrical  method  of  prime 
and  ultimate  ratios  or  by  the  analytical  method 
of  derived  functions,  we  can  use  infinitesimals 
as  a  sure  and  convenient  instrument  for  shortening 
and  simplifying  proofs.  It  is  in  this  manner 
that  we  shorten  the  proofs  of  the  ancients  by  the 
method  of  indivisibles." 

P.  480,  line  3  up  : 

Cf.  on  this  point,  notes  of  mine  on  pp.  106,  108, 
in  Conservation  of  Energy. 

P.  493,  line  9  : 

This  refers  to  Conservation  of  Energy  ^  pp.  51-53, 
86-88;  cf  pp.  94,  97-98. 


loo       THE  SCIENCE  OF  MECHANICS 

P.  499,  line  7  : 

Helmholtz's  famous  memoir  of  1847  is  repro- 
duced, together  with  own  notes  of  188 1,  in  No.  i  of 
OstwalcVs  Klassiker. 

P.  502,  Hne  19  : 

On    this    and    what    follows,    cf.    Conservation  of 
Energy,  pp.  61-64,  69-74,  98-102. 

P.  510,  line  5  : 

The  anecdote  of  Newton  and  the  falling  apple 
rests  on  good  authority.  It  was  stated  to  be  the 
fact  by  Conduitt,  the  husband  of  Newton's  favourite 
niece,  and  was  repeated  later  by  Mrs  Conduitt  to 
Voltaire,  through  whom  it  became  well  known.  It 
was  also  mentioned  by  others,  and  is  confirmed 
by  a  local  tradition.  See  Rouse  Ball,  op.  cit. , 
pp.  11-12;  Rosenberger,  op.  cit.,  pp.  1 19-120. 
Cf.  my  article  in  the  Monist  for  April  19 14, 
vol.  xxiv,  p.  202. 

P.   527,  line  8  up:   "Hamilton's." 

Proc.  of  the  Royal  Irish  Acad.,  March  1847  (read 
1846);  Lectures  on  Quaternions,  Dublin,  1853, 
p.  614;  Elements  of  Quaternions,  London,  1866, 
pp.  100,  718  (a  second  edition  of  the  Elements  was 
published  at  London,  1 899-1 901).  The  concept  of 
Hodograph  was  formed  by  Mobius  as  early  as 
1843  {cf.  his  Mechanik  des  Himmels ;  Werke, 
vol.  iv,  Leipsic,   1887,  pp.  36,  47). 


NOTES  ON  MACH'S  ''MECHANICS''     'oi 

P.  530,  line  4: 

It  is  to  be  remembered  that  Newton  himself, 
although  he  rejected  the  undulatory  theory  of  light 
because  he  did  not  think  that  this  theory  could 
explain  the  rectilinear  propagation  of  light,  con- 
sidered the  whole  of  space  to  be  filled  with  an 
elastic  medium  which  propagates  vibrations  in  a 
manner  analogous  to  that  in  which  the  air  propagates 
vibrations  of  sounds.  The  aether  penetrates  into 
the  pores  of  all  material  bodies  whose  cohesion 
it  brings  about  ;  it  transmits  gravitational  action, 
and  its  irregular  turbulence  constitutes  heat.  Cf. 
E.  T.  Whittaker,  A  History  of  the  Theories  of 
^ther  and  Electricity  from  the  Age  of  Descartes  to 
the  Close  of  the  Nineteenth  Century ^  London  and 
Dublin,  19 10.  Cf.  also  p.  534  of  the  Mechanics, 
and  my  articles  in  the  Monist  for  April  19 14 
(vol.  xxiv,  pp.  219-223)  and  January  and  April 
1915. 

P.   532,  line  3  : 

On  these  investigations  of  Hooke's,  see  my  paper 
in  the  Monist  for  July  19 13. 

P.   554,  line  9: 

On  this  and  other  examples  see  L.  Boltzmann, 
"  Eine  Anfrage  betreffend  ein  Beispiel  zu  Hertz' 
Mechanik,"  faJiresber.  der  dents cJi.  math.-Ver., 
vol.    vii,    1899,   pp.    7^-77* 


I02       THE  SCIENCE  OF  MECHANICS 

P.  55S,  line  13: 

This  article  is  translated  on  pp.  80-85  of  Conser- 
vation of  Energy, 

P.  563,  line  13  : 

This  is  on  pp.  27-28  of  Conservation  of  Energy. 

P.   567,  second  note  : 

See  pp.  75-80,  105,  o{  Conservation  of  Ene^'gy. 

P.  578,  Hne  9  : 

On  Grassmann's  ideas,  cf.  the  references  given  in 
the  above  note  to  p.  480  of  Mechanics. 

P.   580,  line  6  up  : 

See  Consei'vatioyi  of  Energy,  p.  88. 


INDEX 


Abendroth,  W.,  29. 
Absolute  conservation  of  energy, 
44-46. 
space,  time,  and  motion,  x,  xii, 

33-34,  36-44,  49-50- 
Acceleration,     with    Galileo,    21, 

24-26. 
JEihei,  Newton's  views  on,  loi. 
Aime,  67. 
Air,  function  of,  with  Aristotle  and 

Philoponos,  17- 1 8. 
Voltaire's  ideas  on,  16-17. 
Alembert,  J.  L.  d',  see  D'Alembert. 
Ampere,  A.  M.,  69. 
Anding,  A.,  40. 
Archimedes,   i,  2,  3,   16,   19,  57, 

64,  66. 
Archytas,  57. 
Aristotle,    6,   10,   15,   16,   17,   19, 

24,  25,  54,  65,  74. 
Attractions,  Newton's  theorem  on, 

79-80. 
Avenarius,  R.,  x. 
Axes   of  reference,   with   Galileo 

and  Newton,  31,  33-34- 

Baliani,  28. 

Ball,  W.  W.  Rouse,  77,  79,  100. 

Barrow,  I.,  30. 

Beeckmann,  27. 

Benedetti,    G. ,  5,  6,   10,   18,   19, 

26,  27, 
Bernoulli,  Daniel,  66,  68,  83,  84, 
86,  87. 

John,  77,94. 
Bernoullis,  the,  89. 
Bertrand,  Joseph,  72,  74,  98. 

Louis,  84. 
Beuchot,  M.,  16. 
Boltzmann,  L.,  52,  loi. 


Boyle,  R.,  17. 
Brahe,  Tycho,  76. 
Bruno,  Giordano,  18. 

Cardano,  G.,  5,  6,  10,  11,  18,  25, 

27. 
Cauchy,  A.  L.,  67,  95. 
Cavalieri,  B, ,  30. 
Cavendish,  H.,  17. 
Centrifugal    force,    Huygens    on, 

72-73- 
Clairaut,  A.  C,  92,  93. 
Clarke,  Samuel,  29. 
Clebsch,  R.  F.  A.,  71. 
Clifford,  W.  K.,  63. 
Conduitt,  Mrs,  100. 
Copernicus,  23,  33. 
Cornelius,  H.,  x. 
Cotes,  Roger,  80. 
Courtivron,  Marquis  de,  69. 
Cox,  J.,  ix. 
Crew,  H.,  20,  24. 
Crookes,  W.,  50. 

D'Alembert,  J.  L.,  66,  70,  86,  87, 

93.  98. 
D'Alembert's    principle,    70,    87, 

98. 
Dannemann,  P.,  71. 
Darboux,  G.,  98. 
D'Arcy,  P.,  83,  84. 
Delambre,  J.  B.  J.,  64. 
Descartes,   R.,  6,    ii,   12,26,  27, 

30,  66,  67,  73. 
Dingier,  IL,  xi. 
Dorn,  E.,  83. 
Duhem,  P.,  x,  5,  6,  7,  9,   lo,  12, 

14,  15,  24,  26,  28. 
Duhring,  E.,  81,  84. 
I  Dvorak,  V.  50,  51, 

103 


104       THE  SCIENCE  OF  MECHANICS 


Euclid,  3. 

Euler,  L.,  68,  69,  87,  89,  90,  93, 
95>  97. 

Fermat,  P.  de,  30. 
Figure  of  the  earth,  92-93. 
Foncenex,  D.  de,  64. 
FoppI,  44. 

Fourier,  J.  B.  J.,  68,  69. 
Friedlander,  B,  and  J.,  44. 
FuUenius,  B.,  86. 

Galileo,  v,  x,  2,  3,  4,  6,  11,  14, 
16,  17,  18,  19,  20,  21,  22, 
23,  24,  25,    26,  27,  28,    30, 

31.  33,   34,  35,  48,    51,   55, 
64,  65,  66,  67,  71,  74. 

Galileo's  contributions  to  science, 
22. 
precursors,  v,  24-26. 

Gauss,  C.  F.,  52,  6S,  71,  82,  87, 

92,  93,  96. 
Geissler,  II.,  50. 
Gibbs,  J.  W.,  87. 
Gilbert,  16. 

Glaisher,  J.  W.  L.,  79. 
Goldbeck,  E.,  23,  24. 
Grassmann,  H.,  102. 
Gray,  G.  J.,  80. 
Green,  G.,  71,  93. 
Guericke,  O.  von,  17. 
Guldin,  30. 

Haberditzl,  A.,  51. 

Halley,  E.,  77. 

Hamel,  G. ,  xi. 

Hamilton,  Sir  W.  R.,  91,  92. 

Hartmann,  Lieut. -Col.,  49. 

Hausdorff,  F.,  73,  84,  86. 

Hecksher,  A.,  73. 

Helm,  G.,  53. 

Helmholtz,  H.  von,  92,  95,  100. 

Hero,  6,  24,  55,  57. 

Hertz,  H.,  ix,  81. 

Hey  mans,  G.,  x. 

Hipparchus,  19. 

Hodograph,  100. 

Holder,  O.,  x,  I,  92. 

Hollefreund,  K. ,  52. 

Hooke,  R.,  77,  78,  loi. 


Huygens,  C,  73,  74,  75,  84,  85, 

86,  92. 
Hydrodynamics,  93. 
Hydrostatics,  92-93. 
Hypotheses  in  mechanics,  47-48. 

Imagination     with     Hooke     and 

Newton,  77-78. 
Impact,  laws  of,  84-86. 
Inertia,  law  of,  23,  34-44,  82. 
Inertial  systems,  xii,  38-39,  41-42. 
Infinite,  in    mathematics,   Galileo 

on,  24. 
Integration  with  Galileo,  48. 

Jacobi,  C.  G.  J.,  70,  71,  86,  91, 

94,  95- 

Jellett,  J.  II.,  95,  96,  97. 

Jordanus,  Nemorarius,  5,  6,  7,  12. 

Jourdain,  P.  E.  B. ,  vi,  vii,  x,  47, 
52,  64,  69,  75,  78,  81,  82, 
83,    84,    88,  89,  90,  91,  92, 

94,  96,    97,  99,   100,  loi- 

Kantian  traditions,  xi. 
Kasner,  E. ,  24. 
Kehl,  de,  16. 
Kepler,  J.,  30,  44,  76,  77. 
Kirchhoff,  G.,  ix. 
Kleinpeter,  H.,  40. 
Kneller,  Sir  G.,  vi. 
Korn,  A.,  86,  98. 
Korner,  T, ,  97. 

Lacroix,  S.  F.,  95. 
Lagrange,  J.   L.,  64,  65,  66,  67, 
69,    70,    72,  87,  90,  91,  94, 

95,  96,    97,  98,  99- 
Lami,  66. 

Lampa,  A.,  31. 

Lange,  L.,  x,  37,  38,  39,  40,  42. 

Laplace,  P.  S. ,  69.  70. 

Lavoisier,  A.  L.,  17. 

Least  action,  principle  of,  69-70, 

88-92,  93,  96-97- 
constraint,  principle  of,  51-53, 

87-88. 
Leibniz,  G.  W.  von,  30. 
Leonardo,  see  Vinci. 
Lever,  law  of,  1-3. 


INDEX 


105 


Lodge,  O.  J.,  40. 
Love,  A.  E.  H.,  ix,  40. 

M'Cormack,  T.  J.,  v,  vi. 
MacGregor,  J.  G.,  ix,  40,  42. 
Mach,  23,  29,  50,  55,  63,  64,  65, 

66,  75,  81,  87,  88,  89,97. 
Machines  and  tools,  our  heritage 

of,  13.  54-57- 
Maclaurin,  C. ,  98. 
Mansion,  P.,  40. 
Marci,  M.  (del  Monte),  3,  4,  51. 
Mass,  concept  of,  15,  26,  82,  85. 

electromagnetic,  31. 

Mach's  definition  of,  46-47. 

relativity  of,  32-33. 
Maupertuis,  P.  L.  M.  de,  69,  84, 

88,  89,  93,  97. 
Mayer,  A,  ix,  89,  92,  94. 
Mersesne,  27,  65,  66. 
Mobius,  100. 
Moment,  concept  of,  8. 
Momentum,  conservation  of,  83- 

84. 
Motte,  A.,  80. 
Mliller,  A. ,  x. 

Natorp,  P.,  X. 

Nemorarius,  Jordanus,  see  Jor- 
danus. 

Neumann,  C,  37,  50,  52. 

Newton,  vi,  xii,  28,  29,  30,  32, 
33.  34,  35,  36,  Zl.  41,  44, 
45,  46,  49,  55,  66,  73,  74, 
77,  78,  79,  80,  81,  82,  83, 
92,  100,  lOI. 

Newton's  contributions  to  science, 
29-30. 

Nicolas  of  Cusa,  24, 

Nicomedes,  66. 

Nix,  L.,  6. 

Oettingen,  A.  J.  von,  20,  71,  72, 

73,  92,  93- 
Ohm,  M.,  95. 

Ostrogradski,  M.,  68,  91,  94. 
Ostwald,  W.,  52. 

Parallelogram    of  forces,    1 1,   66- 
67. 
of  motions,  23,  65-66. 


Pascal,  B.,  12,  17. 

Pearson,  K. ,  40,  63. 

Pemberton,  II.,  80. 

Perpetual  motion,  impossibility  of, 

9-10,  81. 
Peizoldt,  J.,  X,  39,  40,  54. 
Philoponos,  17,  18. 
Piccolomini,  A.,  26. 
Picot,  C,  27. 
Poincare,  H.,  47,  48. 
Poinsot,  L. ,  69,  70. 
Poisson,  S.  D.,  67. 
Popper,  J.,  ix. 
Potential,  theory  of,  71,  93. 
lever,  Leonardo's,  8,  11. 
Priestley,  J.,  17. 
Projection    with     Benedetti     and 

Galileo,  18-19. 
Proportions  and  equations,  73-74. 

Radiometer,     experiments     with, 

50. 
Reaction-wheel,  experiments  with, 

50-51. 
Richer,  74. 

Roberval,  G.  P.,  6,  12,  30,  65,  66. 
Rodrigues,  O.,  94. 
Rosenberger,  F. ,  30,  82,  loo. 
Rossellini,  56. 

Salviati,  22. 

Salvio,  A.  de,  20,  24. 

Sarpi,  21, 

Scaliger,  J.  C,  26,  27. 

Schmidt,  W.,  6. 

Schuppe,  W. ,  X. 

Schuster,  A.,  50. 

Schiitz,  J.  R.,  44. 

Seeliger,  H.  von,  ix,  39. 

Similarity,  principle  of,  74. 

Simplicius,  25. 

Stackel,  P.,  74,  77,  88,  91,  94. 

Stallo,  J.  B.,  40,  81. 

Stegmann,  95. 

Stevinus,  S.,  2,  3,  4,  6,    11,    15, 

16,  65. 
Strauch,  G.  W.,  95. 
Streintz,  H.,  37. 

Tait,  P.  G.,  41. 
Tartaglia,  N.,  5,  25. 


io6       THE  SCIENCE  OF  MECHANICS 


Taylor's  theorem,  65. 
Theodoras,  56. 
Thomson,  J.,  40,  41. 

W.,41,  47. 
Tides,  Galileo's  theory  of,  31. 
Todhunter,  I.,  74,  92,  93. 
Toricelli,  E.,  25,  67. 

Ubaldi,  G.,  65,  (>-]. 

Vailati,  G.,  x,  i,  2,  15,  28, 
Variation,  concept  of,  94-96. 
Varignon,  P.,  66,  67, 
Vinci,  Leonardo  da,  5,  6,  7,  8,  9, 

10,  II,  18,  25,  26. 
Virtual  displacements,  3-4,  8,  11, 
67-69. 
inequalities  in,  68. 
Vis  viva,  principle  of,  75,  86-87. 
Voider,  B.  de,  86. 


Volkmann,  P.,  x. 
Volta,  17. 

Voltaire,  16,  97,  100. 
Voss,  A.,  66,  67,  68,  69,  74,  82, 
86,  87. 

Wallis,  J.,  30,  66,  67,  73,  84,  85. 
Wangerin,  A.,  71,  93. 
Weber,  H.,  92. 

R.  H.,  92. 
Whittaker,  E.  T.,  loi. 
Wohlwill,  E.,  V,  X,  3,  4,  5,  6,  14, 

17,  18,  20,  21,  23,  27. 
Work,   concept   of,  4,  8,   11,  22, 

Wren,  Sir  C,  'j'],  84. 
Wundt,  W.,  38,  42. 

Zeller,  E.,  6. 
Zemplen,  52. 


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